During the archaeological research in the Middle East you found the traces of three ancient religions: First religion, Second religion and Third religion. You compiled the information on the evolution of each of these beliefs, and you now wonder if the followers of each religion could coexist in peace.

The Word of Universe is a long word containing the lowercase English characters only. At each moment of time, each of the religion beliefs could be described by a word consisting of lowercase English characters.

The three religions can coexist in peace if their descriptions form disjoint subsequences of the Word of Universe. More formally, one can paint some of the characters of the Word of Universe in three colors: 1, 2, 3, so that each character is painted in at most one color, and the description of the i-th religion can be constructed from the Word of Universe by removing all characters that aren't painted in color i.

The religions however evolve. In the beginning, each religion description is empty. Every once in a while, either a character is appended to the end of the description of a single religion, or the last character is dropped from the description. After each change, determine if the religions could coexist in peace.

Input

The first line of the input contains two integers n, q (1 ≤ n ≤ 100 000, 1 ≤ q ≤ 1000) — the length of the Word of Universe and the number of religion evolutions, respectively. The following line contains the Word of Universe — a string of length n consisting of lowercase English characters.

Each of the following line describes a single evolution and is in one of the following formats:

  * + i c (i ∈ \{1, 2, 3\}, c ∈ \{a, b, ..., z\}: append the character c to the end of i-th religion description.
  * - i (i ∈ \{1, 2, 3\}) – remove the last character from the i-th religion description. You can assume that the pattern is non-empty.

You can assume that no religion will have description longer than 250 characters.

Output

Write q lines. The i-th of them should be YES if the religions could coexist in peace after the i-th evolution, or NO otherwise.

You can print each character in any case (either upper or lower).

Examples

Input

6 8
abdabc
+ 1 a
+ 1 d
+ 2 b
+ 2 c
+ 3 a
+ 3 b
+ 1 c
- 2

Output

YES
YES
YES
YES
YES
YES
NO
YES

Input

6 8
abbaab
+ 1 a
+ 2 a
+ 3 a
+ 1 b
+ 2 b
+ 3 b
- 1
+ 2 z

Output

YES
YES
YES
YES
YES
NO
YES
NO

Note

In the first example, after the 6th evolution the religion descriptions are: ad, bc, and ab. The following figure shows how these descriptions form three disjoint subsequences of the Word of Universe:

<image>


Every day Kotlin heroes analyze the statistics of their website. For n days, they wrote out n numbers a_1, a_2, ..., a_n, where a_i is the number of visits on the i-th day.

They believe that a day is bad if there are at least 2 days before it with a strictly greater number of visits. For example, if n=8 and a=[3, 1, 4, 1, 5, 9, 2, 6], then the day 4 is bad (because a_4=1, but there are a_1=3 and a_3=4). Also, the day with the number 7 is bad too.

Write a program that finds the number of bad days.

Input

The first line contains an integer n (1 ≤ n ≤ 2⋅10^5), where n is the number of days. The second line contains n positive integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^9), where a_i is the number of website visits on the i-th day.

Output

Print the number of bad days, i.e. such days that there are at least two days before it with a strictly greater number of visits.

Examples

Input

8
3 1 4 1 5 9 2 6

Output

2

Input

5
1 1 1 1 1

Output

0

Input

13
2 7 1 8 2 8 1 8 2 8 4 5 9

Output

6


After playing Neo in the legendary "Matrix" trilogy, Keanu Reeves started doubting himself: maybe we really live in virtual reality? To find if this is true, he needs to solve the following problem.

Let's call a string consisting of only zeroes and ones good if it contains different numbers of zeroes and ones. For example, 1, 101, 0000 are good, while 01, 1001, and 111000 are not good.

We are given a string s of length n consisting of only zeroes and ones. We need to cut s into minimal possible number of substrings s_1, s_2, …, s_k such that all of them are good. More formally, we have to find minimal by number of strings sequence of good strings s_1, s_2, …, s_k such that their concatenation (joining) equals s, i.e. s_1 + s_2 + ... + s_k = s.

For example, cuttings 110010 into 110 and 010 or into 11 and 0010 are valid, as 110, 010, 11, 0010 are all good, and we can't cut 110010 to the smaller number of substrings as 110010 isn't good itself. At the same time, cutting of 110010 into 1100 and 10 isn't valid as both strings aren't good. Also, cutting of 110010 into 1, 1, 0010 isn't valid, as it isn't minimal, even though all 3 strings are good.

Can you help Keanu? We can show that the solution always exists. If there are multiple optimal answers, print any.

Input

The first line of the input contains a single integer n (1≤ n ≤ 100) — the length of the string s.

The second line contains the string s of length n consisting only from zeros and ones.

Output

In the first line, output a single integer k (1≤ k) — a minimal number of strings you have cut s into.

In the second line, output k strings s_1, s_2, …, s_k separated with spaces. The length of each string has to be positive. Their concatenation has to be equal to s and all of them have to be good.

If there are multiple answers, print any.

Examples

Input

1
1

Output

1
1

Input

2
10

Output

2
1 0

Input

6
100011

Output

2
100 011

Note

In the first example, the string 1 wasn't cut at all. As it is good, the condition is satisfied.

In the second example, 1 and 0 both are good. As 10 isn't good, the answer is indeed minimal.

In the third example, 100 and 011 both are good. As 100011 isn't good, the answer is indeed minimal.


Cengiz recently learned Fibonacci numbers and now he is studying different algorithms to find them. After getting bored of reading them, he came with his own new type of numbers that he named XORinacci numbers. He defined them as follows:

  * f(0) = a;
  * f(1) = b;
  * f(n) = f(n-1) ⊕ f(n-2) when n > 1, where ⊕ denotes the [bitwise XOR operation](https://en.wikipedia.org/wiki/Bitwise_operation#XOR).

You are given three integers a, b, and n, calculate f(n).

You have to answer for T independent test cases.

Input

The input contains one or more independent test cases.

The first line of input contains a single integer T (1 ≤ T ≤ 10^3), the number of test cases.

Each of the T following lines contains three space-separated integers a, b, and n (0 ≤ a, b, n ≤ 10^9) respectively.

Output

For each test case, output f(n).

Example

Input

3
3 4 2
4 5 0
325 265 1231232

Output

7
4
76

Note

In the first example, f(2) = f(0) ⊕ f(1) = 3 ⊕ 4 = 7.


Your math teacher gave you the following problem:

There are n segments on the x-axis, [l_1; r_1], [l_2; r_2], …, [l_n; r_n]. The segment [l; r] includes the bounds, i.e. it is a set of such x that l ≤ x ≤ r. The length of the segment [l; r] is equal to r - l.

Two segments [a; b] and [c; d] have a common point (intersect) if there exists x that a ≤ x ≤ b and c ≤ x ≤ d. For example, [2; 5] and [3; 10] have a common point, but [5; 6] and [1; 4] don't have.

You should add one segment, which has at least one common point with each of the given segments and as short as possible (i.e. has minimal length). The required segment can degenerate to be a point (i.e a segment with length zero). The added segment may or may not be among the given n segments.

In other words, you need to find a segment [a; b], such that [a; b] and every [l_i; r_i] have a common point for each i, and b-a is minimal.

Input

The first line contains integer number t (1 ≤ t ≤ 100) — the number of test cases in the input. Then t test cases follow.

The first line of each test case contains one integer n (1 ≤ n ≤ 10^{5}) — the number of segments. The following n lines contain segment descriptions: the i-th of them contains two integers l_i,r_i (1 ≤ l_i ≤ r_i ≤ 10^{9}).

The sum of all values n over all the test cases in the input doesn't exceed 10^5.

Output

For each test case, output one integer — the smallest possible length of the segment which has at least one common point with all given segments.

Example

Input

4
3
4 5
5 9
7 7
5
11 19
4 17
16 16
3 12
14 17
1
1 10
1
1 1

Output

2
4
0
0

Note

In the first test case of the example, we can choose the segment [5;7] as the answer. It is the shortest segment that has at least one common point with all given segments.


You are given an infinite checkered field. You should get from a square (x1; y1) to a square (x2; y2). Using the shortest path is not necessary. You can move on the field squares in four directions. That is, when you are positioned in any square, you can move to any other side-neighboring one.

A square (x; y) is considered bad, if at least one of the two conditions is fulfilled:

  * |x + y| ≡ 0 (mod 2a),
  * |x - y| ≡ 0 (mod 2b).

Your task is to find the minimum number of bad cells one will have to visit on the way from (x1; y1) to (x2; y2).

Input

The only line contains integers a, b, x1, y1, x2 and y2 — the parameters of the bad squares, the coordinates of the initial and the final squares correspondingly (2 ≤ a, b ≤ 109 and |x1|,|y1|,|x2|,|y2| ≤ 109). It is guaranteed that the initial and the final square aren't bad.

Output

Print a single number — the minimum number of bad cells that one will have to visit in order to travel from square (x1; y1) to square (x2; y2).

Examples

Input

2 2 1 0 0 1

Output

1

Input

2 2 10 11 0 1

Output

5

Input

2 4 3 -1 3 7

Output

2

Note

In the third sample one of the possible paths in (3;-1)->(3;0)->(3;1)->(3;2)->(4;2)->(4;3)->(4;4)->(4;5)->(4;6)->(4;7)->(3;7). Squares (3;1) and (4;4) are bad.


You are given a positive integer m and two integer sequence: a=[a_1, a_2, …, a_n] and b=[b_1, b_2, …, b_n]. Both of these sequence have a length n.

Permutation is a sequence of n different positive integers from 1 to n. For example, these sequences are permutations: [1], [1,2], [2,1], [6,7,3,4,1,2,5]. These are not: [0], [1,1], [2,3].

You need to find the non-negative integer x, and increase all elements of a_i by x, modulo m (i.e. you want to change a_i to (a_i + x) mod m), so it would be possible to rearrange elements of a to make it equal b, among them you need to find the smallest possible x.

In other words, you need to find the smallest non-negative integer x, for which it is possible to find some permutation p=[p_1, p_2, …, p_n], such that for all 1 ≤ i ≤ n, (a_i + x) mod m = b_{p_i}, where y mod m — remainder of division of y by m.

For example, if m=3, a = [0, 0, 2, 1], b = [2, 0, 1, 1], you can choose x=1, and a will be equal to [1, 1, 0, 2] and you can rearrange it to make it equal [2, 0, 1, 1], which is equal to b.

Input

The first line contains two integers n,m (1 ≤ n ≤ 2000, 1 ≤ m ≤ 10^9): number of elemens in arrays and m.

The second line contains n integers a_1, a_2, …, a_n (0 ≤ a_i < m).

The third line contains n integers b_1, b_2, …, b_n (0 ≤ b_i < m).

It is guaranteed that there exists some non-negative integer x, such that it would be possible to find some permutation p_1, p_2, …, p_n such that (a_i + x) mod m = b_{p_i}.

Output

Print one integer, the smallest non-negative integer x, such that it would be possible to find some permutation p_1, p_2, …, p_n such that (a_i + x) mod m = b_{p_i} for all 1 ≤ i ≤ n.

Examples

Input

4 3
0 0 2 1
2 0 1 1

Output

1

Input

3 2
0 0 0
1 1 1

Output

1

Input

5 10
0 0 0 1 2
2 1 0 0 0

Output

0


You're given an array a_1, …, a_n of n non-negative integers.

Let's call it sharpened if and only if there exists an integer 1 ≤ k ≤ n such that a_1 < a_2 < … < a_k and a_k > a_{k+1} > … > a_n. In particular, any strictly increasing or strictly decreasing array is sharpened. For example:

  * The arrays [4], [0, 1], [12, 10, 8] and [3, 11, 15, 9, 7, 4] are sharpened;
  * The arrays [2, 8, 2, 8, 6, 5], [0, 1, 1, 0] and [2, 5, 6, 9, 8, 8] are not sharpened.

You can do the following operation as many times as you want: choose any strictly positive element of the array, and decrease it by one. Formally, you can choose any i (1 ≤ i ≤ n) such that a_i>0 and assign a_i := a_i - 1.

Tell if it's possible to make the given array sharpened using some number (possibly zero) of these operations.

Input

The input consists of multiple test cases. The first line contains a single integer t (1 ≤ t ≤ 15\ 000) — the number of test cases. The description of the test cases follows.

The first line of each test case contains a single integer n (1 ≤ n ≤ 3 ⋅ 10^5).

The second line of each test case contains a sequence of n non-negative integers a_1, …, a_n (0 ≤ a_i ≤ 10^9).

It is guaranteed that the sum of n over all test cases does not exceed 3 ⋅ 10^5.

Output

For each test case, output a single line containing "Yes" (without quotes) if it's possible to make the given array sharpened using the described operations, or "No" (without quotes) otherwise.

Example

Input

10
1
248618
3
12 10 8
6
100 11 15 9 7 8
4
0 1 1 0
2
0 0
2
0 1
2
1 0
2
1 1
3
0 1 0
3
1 0 1

Output

Yes
Yes
Yes
No
No
Yes
Yes
Yes
Yes
No

Note

In the first and the second test case of the first test, the given array is already sharpened.

In the third test case of the first test, we can transform the array into [3, 11, 15, 9, 7, 4] (decrease the first element 97 times and decrease the last element 4 times). It is sharpened because 3 < 11 < 15 and 15 > 9 > 7 > 4.

In the fourth test case of the first test, it's impossible to make the given array sharpened.


You want to perform the combo on your opponent in one popular fighting game. The combo is the string s consisting of n lowercase Latin letters. To perform the combo, you have to press all buttons in the order they appear in s. I.e. if s="abca" then you have to press 'a', then 'b', 'c' and 'a' again.

You know that you will spend m wrong tries to perform the combo and during the i-th try you will make a mistake right after p_i-th button (1 ≤ p_i < n) (i.e. you will press first p_i buttons right and start performing the combo from the beginning). It is guaranteed that during the m+1-th try you press all buttons right and finally perform the combo.

I.e. if s="abca", m=2 and p = [1, 3] then the sequence of pressed buttons will be 'a' (here you're making a mistake and start performing the combo from the beginning), 'a', 'b', 'c', (here you're making a mistake and start performing the combo from the beginning), 'a' (note that at this point you will not perform the combo because of the mistake), 'b', 'c', 'a'.

Your task is to calculate for each button (letter) the number of times you'll press it.

You have to answer t independent test cases.

Input

The first line of the input contains one integer t (1 ≤ t ≤ 10^4) — the number of test cases.

Then t test cases follow.

The first line of each test case contains two integers n and m (2 ≤ n ≤ 2 ⋅ 10^5, 1 ≤ m ≤ 2 ⋅ 10^5) — the length of s and the number of tries correspondingly.

The second line of each test case contains the string s consisting of n lowercase Latin letters.

The third line of each test case contains m integers p_1, p_2, ..., p_m (1 ≤ p_i < n) — the number of characters pressed right during the i-th try.

It is guaranteed that the sum of n and the sum of m both does not exceed 2 ⋅ 10^5 (∑ n ≤ 2 ⋅ 10^5, ∑ m ≤ 2 ⋅ 10^5).

It is guaranteed that the answer for each letter does not exceed 2 ⋅ 10^9.

Output

For each test case, print the answer — 26 integers: the number of times you press the button 'a', the number of times you press the button 'b', ..., the number of times you press the button 'z'.

Example

Input

3
4 2
abca
1 3
10 5
codeforces
2 8 3 2 9
26 10
qwertyuioplkjhgfdsazxcvbnm
20 10 1 2 3 5 10 5 9 4

Output

4 2 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
0 0 9 4 5 3 0 0 0 0 0 0 0 0 9 0 0 3 1 0 0 0 0 0 0 0
2 1 1 2 9 2 2 2 5 2 2 2 1 1 5 4 11 8 2 7 5 1 10 1 5 2

Note

The first test case is described in the problem statement. Wrong tries are "a", "abc" and the final try is "abca". The number of times you press 'a' is 4, 'b' is 2 and 'c' is 2.

In the second test case, there are five wrong tries: "co", "codeforc", "cod", "co", "codeforce" and the final try is "codeforces". The number of times you press 'c' is 9, 'd' is 4, 'e' is 5, 'f' is 3, 'o' is 9, 'r' is 3 and 's' is 1.


Polycarp has recently created a new level in this cool new game Berlio Maker 85 and uploaded it online. Now players from all over the world can try his level.

All levels in this game have two stats to them: the number of plays and the number of clears. So when a player attempts the level, the number of plays increases by 1. If he manages to finish the level successfully then the number of clears increases by 1 as well. Note that both of the statistics update at the same time (so if the player finishes the level successfully then the number of plays will increase at the same time as the number of clears).

Polycarp is very excited about his level, so he keeps peeking at the stats to know how hard his level turns out to be.

So he peeked at the stats n times and wrote down n pairs of integers — (p_1, c_1), (p_2, c_2), ..., (p_n, c_n), where p_i is the number of plays at the i-th moment of time and c_i is the number of clears at the same moment of time. The stats are given in chronological order (i.e. the order of given pairs is exactly the same as Polycarp has written down).

Between two consecutive moments of time Polycarp peeked at the stats many players (but possibly zero) could attempt the level.

Finally, Polycarp wonders if he hasn't messed up any records and all the pairs are correct. If there could exist such a sequence of plays (and clears, respectively) that the stats were exactly as Polycarp has written down, then he considers his records correct.

Help him to check the correctness of his records.

For your convenience you have to answer multiple independent test cases.

Input

The first line contains a single integer T (1 ≤ T ≤ 500) — the number of test cases.

The first line of each test case contains a single integer n (1 ≤ n ≤ 100) — the number of moments of time Polycarp peeked at the stats.

Each of the next n lines contains two integers p_i and c_i (0 ≤ p_i, c_i ≤ 1000) — the number of plays and the number of clears of the level at the i-th moment of time.

Note that the stats are given in chronological order.

Output

For each test case print a single line.

If there could exist such a sequence of plays (and clears, respectively) that the stats were exactly as Polycarp has written down, then print "YES".

Otherwise, print "NO".

You can print each letter in any case (upper or lower).

Example

Input

6
3
0 0
1 1
1 2
2
1 0
1000 3
4
10 1
15 2
10 2
15 2
1
765 432
2
4 4
4 3
5
0 0
1 0
1 0
1 0
1 0

Output

NO
YES
NO
YES
NO
YES

Note

In the first test case at the third moment of time the number of clears increased but the number of plays did not, that couldn't have happened.

The second test case is a nice example of a Super Expert level.

In the third test case the number of plays decreased, which is impossible.

The fourth test case is probably an auto level with a single jump over the spike.

In the fifth test case the number of clears decreased, which is also impossible.

Nobody wanted to play the sixth test case; Polycarp's mom attempted it to make him feel better, however, she couldn't clear it.


The statement of this problem is the same as the statement of problem C1. The only difference is that, in problem C1, n is always even, and in C2, n is always odd.

You are given a regular polygon with 2 ⋅ n vertices (it's convex and has equal sides and equal angles) and all its sides have length 1. Let's name it as 2n-gon.

Your task is to find the square of the minimum size such that you can embed 2n-gon in the square. Embedding 2n-gon in the square means that you need to place 2n-gon in the square in such way that each point which lies inside or on a border of 2n-gon should also lie inside or on a border of the square.

You can rotate 2n-gon and/or the square.

Input

The first line contains a single integer T (1 ≤ T ≤ 200) — the number of test cases.

Next T lines contain descriptions of test cases — one per line. Each line contains single odd integer n (3 ≤ n ≤ 199). Don't forget you need to embed 2n-gon, not an n-gon.

Output

Print T real numbers — one per test case. For each test case, print the minimum length of a side of the square 2n-gon can be embedded in. Your answer will be considered correct if its absolute or relative error doesn't exceed 10^{-6}.

Example

Input

3
3
5
199

Output

1.931851653
3.196226611
126.687663595


You are given an integer n. In one move, you can either multiply n by two or divide n by 6 (if it is divisible by 6 without the remainder).

Your task is to find the minimum number of moves needed to obtain 1 from n or determine if it's impossible to do that.

You have to answer t independent test cases.

Input

The first line of the input contains one integer t (1 ≤ t ≤ 2 ⋅ 10^4) — the number of test cases. Then t test cases follow.

The only line of the test case contains one integer n (1 ≤ n ≤ 10^9).

Output

For each test case, print the answer — the minimum number of moves needed to obtain 1 from n if it's possible to do that or -1 if it's impossible to obtain 1 from n.

Example

Input

7
1
2
3
12
12345
15116544
387420489

Output

0
-1
2
-1
-1
12
36

Note

Consider the sixth test case of the example. The answer can be obtained by the following sequence of moves from the given integer 15116544:

  1. Divide by 6 and get 2519424;
  2. divide by 6 and get 419904;
  3. divide by 6 and get 69984;
  4. divide by 6 and get 11664;
  5. multiply by 2 and get 23328;
  6. divide by 6 and get 3888;
  7. divide by 6 and get 648;
  8. divide by 6 and get 108;
  9. multiply by 2 and get 216;
  10. divide by 6 and get 36;
  11. divide by 6 and get 6;
  12. divide by 6 and get 1.


You are given an array a of n integers.

You want to make all elements of a equal to zero by doing the following operation exactly three times:

  * Select a segment, for each number in this segment we can add a multiple of len to it, where len is the length of this segment (added integers can be different).

It can be proven that it is always possible to make all elements of a equal to zero.

Input

The first line contains one integer n (1 ≤ n ≤ 100 000): the number of elements of the array.

The second line contains n elements of an array a separated by spaces: a_1, a_2, ..., a_n (-10^9 ≤ a_i ≤ 10^9).

Output

The output should contain six lines representing three operations.

For each operation, print two lines:

  * The first line contains two integers l, r (1 ≤ l ≤ r ≤ n): the bounds of the selected segment.

  * The second line contains r-l+1 integers b_l, b_{l+1}, ..., b_r (-10^{18} ≤ b_i ≤ 10^{18}): the numbers to add to a_l, a_{l+1}, …, a_r, respectively; b_i should be divisible by r - l + 1.

Example

Input

4
1 3 2 4

Output

1 1
-1
3 4
4 2
2 4
-3 -6 -6


Ori and Sein have overcome many difficult challenges. They finally lit the Shrouded Lantern and found Gumon Seal, the key to the Forlorn Ruins. When they tried to open the door to the ruins... nothing happened.

Ori was very surprised, but Sein gave the explanation quickly: clever Gumon decided to make an additional defence for the door.

There are n lamps with Spirit Tree's light. Sein knows the time of turning on and off for the i-th lamp — l_i and r_i respectively. To open the door you have to choose k lamps in such a way that there will be a moment of time when they all will be turned on.

While Sein decides which of the k lamps to pick, Ori is interested: how many ways there are to pick such k lamps that the door will open? It may happen that Sein may be wrong and there are no such k lamps. The answer might be large, so print it modulo 998 244 353.

Input

First line contains two integers n and k (1 ≤ n ≤ 3 ⋅ 10^5, 1 ≤ k ≤ n) — total number of lamps and the number of lamps that must be turned on simultaneously.

Next n lines contain two integers l_i ans r_i (1 ≤ l_i ≤ r_i ≤ 10^9) — period of time when i-th lamp is turned on.

Output

Print one integer — the answer to the task modulo 998 244 353.

Examples

Input

7 3
1 7
3 8
4 5
6 7
1 3
5 10
8 9

Output

9

Input

3 1
1 1
2 2
3 3

Output

3

Input

3 2
1 1
2 2
3 3

Output

0

Input

3 3
1 3
2 3
3 3

Output

1

Input

5 2
1 3
2 4
3 5
4 6
5 7

Output

7

Note

In first test case there are nine sets of k lamps: (1, 2, 3), (1, 2, 4), (1, 2, 5), (1, 2, 6), (1, 3, 6), (1, 4, 6), (2, 3, 6), (2, 4, 6), (2, 6, 7).

In second test case k=1, so the answer is 3.

In third test case there are no such pairs of lamps.

In forth test case all lamps are turned on in a time 3, so the answer is 1.

In fifth test case there are seven sets of k lamps: (1, 2), (1, 3), (2, 3), (2, 4), (3, 4), (3, 5), (4, 5).


You are given an array a_1, a_2, …, a_n of integers. This array is non-increasing.

Let's consider a line with n shops. The shops are numbered with integers from 1 to n from left to right. The cost of a meal in the i-th shop is equal to a_i.

You should process q queries of two types:

  * 1 x y: for each shop 1 ≤ i ≤ x set a_{i} = max(a_{i}, y).
  * 2 x y: let's consider a hungry man with y money. He visits the shops from x-th shop to n-th and if he can buy a meal in the current shop he buys one item of it. Find how many meals he will purchase. The man can buy a meal in the shop i if he has at least a_i money, and after it his money decreases by a_i.

Input

The first line contains two integers n, q (1 ≤ n, q ≤ 2 ⋅ 10^5).

The second line contains n integers a_{1},a_{2}, …, a_{n} (1 ≤ a_{i} ≤ 10^9) — the costs of the meals. It is guaranteed, that a_1 ≥ a_2 ≥ … ≥ a_n.

Each of the next q lines contains three integers t, x, y (1 ≤ t ≤ 2, 1≤ x ≤ n, 1 ≤ y ≤ 10^9), each describing the next query.

It is guaranteed that there exists at least one query of type 2.

Output

For each query of type 2 output the answer on the new line.

Example

Input

10 6
10 10 10 6 6 5 5 5 3 1
2 3 50
2 4 10
1 3 10
2 2 36
1 4 7
2 2 17

Output

8
3
6
2

Note

In the first query a hungry man will buy meals in all shops from 3 to 10.

In the second query a hungry man will buy meals in shops 4, 9, and 10.

After the third query the array a_1, a_2, …, a_n of costs won't change and will be \{10, 10, 10, 6, 6, 5, 5, 5, 3, 1\}.

In the fourth query a hungry man will buy meals in shops 2, 3, 4, 5, 9, and 10.

After the fifth query the array a of costs will be \{10, 10, 10, 7, 6, 5, 5, 5, 3, 1\}.

In the sixth query a hungry man will buy meals in shops 2 and 4.


After his wife's tragic death, Eurydice, Orpheus descended to the realm of death to see her. Reaching its gates was uneasy, but passing through them proved to be even more challenging. Mostly because of Cerberus, the three-headed hound of Hades.

Orpheus, a famous poet, and musician plans to calm Cerberus with his poetry and safely walk past him. He created a very peculiar poem for Cerberus. It consists only of lowercase English letters.

We call a poem's substring a palindrome if and only if it reads the same backwards and forwards. A string a is a substring of a string b if a can be obtained from b by deleting several (possibly zero or all) characters from the beginning and several (possibly zero or all) characters from the end.

Unfortunately, Cerberus dislikes palindromes of length greater than 1. For example in the poem abaa the hound of Hades wouldn't like substrings aba and aa.

Orpheus can only calm Cerberus if the hound likes his poetry. That's why he wants to change his poem so that it does not contain any palindrome substrings of length greater than 1.

Orpheus can modify the poem by replacing a letter at any position with any lowercase English letter. He can use this operation arbitrarily many times (possibly zero). Since there can be many palindromes in his poem, he may have to make some corrections. But how many, exactly? Given the poem, determine the minimal number of letters that have to be changed so that the poem does not contain any palindromes of length greater than 1.

Input

The first line of the input contains a single integer t (1 ≤ t ≤ 10^5) denoting the number of test cases, then t test cases follow.

The first and only line of each test case contains a non-empty string of lowercase English letters, Orpheus' poem.

The sum of the length of Orpheus' poems in all test cases will not exceed 10^5.

Output

You should output t lines, i-th line should contain a single integer, answer to the i-th test case.

Example

Input

7
babba
abaac
codeforces
zeroorez
abcdcba
bbbbbbb
a

Output

1
1
0
1
1
4
0

Note

In the first test case, we can replace the third character with c and obtain a palindrome-less poem bacba.

In the second test case, we can replace the third character with d and obtain a palindrome-less poem abdac.

In the third test case, the initial poem already doesn't contain any palindromes, so Orpheus doesn't need to change anything there.


A permutation — is a sequence of length n integers from 1 to n, in which all the numbers occur exactly once. For example, [1], [3, 5, 2, 1, 4], [1, 3, 2] — permutations, and [2, 3, 2], [4, 3, 1], [0] — no.

Polycarp was recently gifted a permutation a[1 ... n] of length n. Polycarp likes trees more than permutations, so he wants to transform permutation a into a rooted binary tree. He transforms an array of different integers into a tree as follows:

  * the maximum element of the array becomes the root of the tree;
  * all elements to the left of the maximum — form a left subtree (which is built according to the same rules but applied to the left part of the array), but if there are no elements to the left of the maximum, then the root has no left child;
  * all elements to the right of the maximum — form a right subtree (which is built according to the same rules but applied to the right side of the array), but if there are no elements to the right of the maximum, then the root has no right child.

For example, if he builds a tree by permutation a=[3, 5, 2, 1, 4], then the root will be the element a_2=5, and the left subtree will be the tree that will be built for the subarray a[1 ... 1] = [3], and the right one — for the subarray a[3 ... 5] = [2, 1, 4]. As a result, the following tree will be built:

<image> The tree corresponding to the permutation a=[3, 5, 2, 1, 4].

Another example: let the permutation be a=[1, 3, 2, 7, 5, 6, 4]. In this case, the tree looks like this:

<image> The tree corresponding to the permutation a=[1, 3, 2, 7, 5, 6, 4].

Let us denote by d_v the depth of the vertex a_v, that is, the number of edges on the path from the root to the vertex numbered a_v. Note that the root depth is zero. Given the permutation a, for each vertex, find the value of d_v.

Input

The first line contains one integer t (1 ≤ t ≤ 100) — the number of test cases. Then t test cases follow.

The first line of each test case contains an integer n (1 ≤ n ≤ 100) — the length of the permutation.

This is followed by n numbers a_1, a_2, …, a_n — permutation a.

Output

For each test case, output n values — d_1, d_2, …, d_n.

Example

Input

3
5
3 5 2 1 4
1
1
4
4 3 1 2

Output

1 0 2 3 1
0
0 1 3 2


Let us denote by d(n) the sum of all divisors of the number n, i.e. d(n) = ∑_{k | n} k.

For example, d(1) = 1, d(4) = 1+2+4=7, d(6) = 1+2+3+6=12.

For a given number c, find the minimum n such that d(n) = c.

Input

The first line contains one integer t (1 ≤ t ≤ 10^4). Then t test cases follow.

Each test case is characterized by one integer c (1 ≤ c ≤ 10^7).

Output

For each test case, output:

  * "-1" if there is no such n that d(n) = c;
  * n, otherwise.

Example

Input

12
1
2
3
4
5
6
7
8
9
10
39
691

Output

1
-1
2
3
-1
5
4
7
-1
-1
18
-1


You are given a multiset of points on the plane with integer coordinates. Find the maximum distance between two points from this multiset.

Input

The first line of input contains the number of points n (2 ≤ n ≤ 50). Next, n pairs of lines follow, each describing a single point: the first line contains x-coordinate, the second one — the y-coordinate ( - 50 ≤ x, y ≤ 50). Some of the points can have identical coordinates.

Output

Output the maximum distance between two points from this multiset. The answer is considered to be correct if its absolute or relative error does not exceed 10 - 4.

Examples

Input

3
0
1
2
3
4
5

Output

5.656854249

Input

3
10
12
-5
8
10
12

Output

15.5241747

Note

In the first case the maximum distance is between points (0, 1) and (4, 5). In the second case two of the points are the same, so the maximum distance is between one of them and the third point.


In some country live wizards. They love playing with numbers.

The blackboard has two numbers written on it — a and b. The order of the numbers is not important. Let's consider a ≤ b for the sake of definiteness. The players can cast one of the two spells in turns:

  * Replace b with b - ak. Number k can be chosen by the player, considering the limitations that k > 0 and b - ak ≥ 0. Number k is chosen independently each time an active player casts a spell.
  * Replace b with b mod a.

If a > b, similar moves are possible.

If at least one of the numbers equals zero, a player can't make a move, because taking a remainder modulo zero is considered somewhat uncivilized, and it is far too boring to subtract a zero. The player who cannot make a move, loses.

To perform well in the magic totalizator, you need to learn to quickly determine which player wins, if both wizards play optimally: the one that moves first or the one that moves second.

Input

The first line contains a single integer t — the number of input data sets (1 ≤ t ≤ 104). Each of the next t lines contains two integers a, b (0 ≤ a, b ≤ 1018). The numbers are separated by a space.

Please do not use the %lld specificator to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specificator.

Output

For any of the t input sets print "First" (without the quotes) if the player who moves first wins. Print "Second" (without the quotes) if the player who moves second wins. Print the answers to different data sets on different lines in the order in which they are given in the input.

Examples

Input

4
10 21
31 10
0 1
10 30

Output

First
Second
Second
First

Note

In the first sample, the first player should go to (11,10). Then, after a single move of the second player to (1,10), he will take 10 modulo 1 and win.

In the second sample the first player has two moves to (1,10) and (21,10). After both moves the second player can win.

In the third sample, the first player has no moves.

In the fourth sample, the first player wins in one move, taking 30 modulo 10.


You are given a number n. Print n lines, i-th line should consist of i characters "*". Lines' indices are 1-based.

Input

The only line of input contains an integer n (1 ≤ n ≤ 50).

Output

Output the described pattern.

Examples

Input

3

Output

*
**
***

Input

6

Output

*
**
***
****
*****
******


Flatland is inhabited by pixels of three colors: red, green and blue. We know that if two pixels of different colors meet in a violent fight, only one of them survives the fight (that is, the total number of pixels decreases by one). Besides, if pixels of colors x and y (x ≠ y) meet in a violent fight, then the pixel that survives the fight immediately changes its color to z (z ≠ x; z ≠ y). Pixels of the same color are friends, so they don't fight.

The King of Flatland knows that his land will be peaceful and prosperous when the pixels are of the same color. For each of the three colors you know the number of pixels of this color that inhabit Flatland. Help the king and determine whether fights can bring peace and prosperity to the country and if it is possible, find the minimum number of fights needed to make the land peaceful and prosperous.

Input

The first line contains three space-separated integers a, b and c (0 ≤ a, b, c ≤ 231; a + b + c > 0) — the number of red, green and blue pixels, correspondingly.

Output

Print a single number — the minimum number of pixel fights before the country becomes peaceful and prosperous. If making the country peaceful and prosperous is impossible, print -1.

Examples

Input

1 1 1

Output

1

Input

3 1 0

Output

3

Note

In the first test sample the country needs only one fight to achieve peace and prosperity. Besides, it can be any fight whatsoever. For example, let's assume that the green and the blue pixels fight, then the surviving pixel will be red. As a result, after the fight there are two red pixels. There won't be other pixels.

In the second sample the following sequence of fights is possible: red and blue, green and red, red and blue. As a result, after all fights there is one green pixel left.


Some days ago, I learned the concept of LCM (least common multiple). I've played with it for several times and I want to make a big number with it.

But I also don't want to use many numbers, so I'll choose three positive integers (they don't have to be distinct) which are not greater than n. Can you help me to find the maximum possible least common multiple of these three integers?

Input

The first line contains an integer n (1 ≤ n ≤ 106) — the n mentioned in the statement.

Output

Print a single integer — the maximum possible LCM of three not necessarily distinct positive integers that are not greater than n.

Examples

Input

9

Output

504

Input

7

Output

210

Note

The least common multiple of some positive integers is the least positive integer which is multiple for each of them.

The result may become very large, 32-bit integer won't be enough. So using 64-bit integers is recommended.

For the last example, we can chose numbers 7, 6, 5 and the LCM of them is 7·6·5 = 210. It is the maximum value we can get.


Berland Government decided to improve relations with neighboring countries. First of all, it was decided to build new roads so that from each city of Berland and neighboring countries it became possible to reach all the others. There are n cities in Berland and neighboring countries in total and exactly n - 1 two-way roads. Because of the recent financial crisis, the Berland Government is strongly pressed for money, so to build a new road it has to close some of the existing ones. Every day it is possible to close one existing road and immediately build a new one. Your task is to determine how many days would be needed to rebuild roads so that from each city it became possible to reach all the others, and to draw a plan of closure of old roads and building of new ones.

Input

The first line contains integer n (2 ≤ n ≤ 1000) — amount of cities in Berland and neighboring countries. Next n - 1 lines contain the description of roads. Each road is described by two space-separated integers ai, bi (1 ≤ ai, bi ≤ n, ai ≠ bi) — pair of cities, which the road connects. It can't be more than one road between a pair of cities. No road connects the city with itself.

Output

Output the answer, number t — what is the least amount of days needed to rebuild roads so that from each city it became possible to reach all the others. Then output t lines — the plan of closure of old roads and building of new ones. Each line should describe one day in the format i j u v — it means that road between cities i and j became closed and a new road between cities u and v is built. Cities are numbered from 1. If the answer is not unique, output any.

Examples

Input

2
1 2

Output

0

Input

7
1 2
2 3
3 1
4 5
5 6
6 7

Output

1
3 1 3 7


Bessie and the cows have recently been playing with "cool" sequences and are trying to construct some. Unfortunately they are bad at arithmetic, so they need your help!

A pair (x, y) of positive integers is "cool" if x can be expressed as the sum of y consecutive integers (not necessarily positive). A sequence (a1, a2, ..., an) is "cool" if the pairs (a1, a2), (a2, a3), ..., (an - 1, an) are all cool.

The cows have a sequence of n positive integers, a1, a2, ..., an. In one move, they may replace some ai with any other positive integer (there are no other limits on the new value of ai). Determine the smallest number of moves needed to make the resulting sequence cool.

Input

The first line contains a single integer, n (2 ≤ n ≤ 5000). The next line contains n space-separated integers, a1, a2, ..., an (1 ≤ ai ≤ 1015).

Please do not use the %lld specifier to read or write 64-bit integers in С++. It is preferred to use the cin, cout streams or the %I64d specifier.

Output

A single integer, the minimum number of ai that must be changed to make the sequence cool.

Examples

Input

3
6 4 1

Output

0

Input

4
20 6 3 4

Output

2

Note

In the first sample, the sequence is already cool, so we don't need to change any elements. In the second sample, we can change a2 to 5 and a3 to 10 to make (20, 5, 10, 4) which is cool. This changes 2 elements.


Polycarpus is sure that his life fits the description: "first there is a white stripe, then a black one, then a white one again". So, Polycarpus is sure that this rule is going to fulfill during the next n days. Polycarpus knows that he is in for w good events and b not-so-good events. At least one event is going to take place during each day. As each day is unequivocally characterizes as a part of a white or a black stripe, then each day is going to have events of the same type only (ether good or not-so-good).

What is the number of distinct ways this scenario can develop over the next n days if Polycarpus is in for a white stripe (a stripe that has good events only, the stripe's length is at least 1 day), the a black stripe (a stripe that has not-so-good events only, the stripe's length is at least 1 day) and a white stripe again (a stripe that has good events only, the stripe's length is at least 1 day). Each of n days will belong to one of the three stripes only.

Note that even the events of the same type are distinct from each other. Even if some events occur on the same day, they go in some order (there are no simultaneous events).

Write a code that prints the number of possible configurations to sort the events into days. See the samples for clarifications on which scenarios should be considered distinct. Print the answer modulo 1000000009 (109 + 9).

Input

The single line of the input contains integers n, w and b (3 ≤ n ≤ 4000, 2 ≤ w ≤ 4000, 1 ≤ b ≤ 4000) — the number of days, the number of good events and the number of not-so-good events. It is guaranteed that w + b ≥ n.

Output

Print the required number of ways modulo 1000000009 (109 + 9).

Examples

Input

3 2 1

Output

2

Input

4 2 2

Output

4

Input

3 2 2

Output

4

Note

We'll represent the good events by numbers starting from 1 and the not-so-good events — by letters starting from 'a'. Vertical lines separate days.

In the first sample the possible ways are: "1|a|2" and "2|a|1". In the second sample the possible ways are: "1|a|b|2", "2|a|b|1", "1|b|a|2" and "2|b|a|1". In the third sample the possible ways are: "1|ab|2", "2|ab|1", "1|ba|2" and "2|ba|1".


Don't put up with what you're sick of! The Smart Beaver decided to escape from the campus of Beaver Science Academy (BSA). BSA is a b × b square on a plane. Each point x, y (0 ≤ x, y ≤ b) belongs to BSA. To make the path quick and funny, the Beaver constructed a Beaveractor, an effective and comfortable types of transport.

The campus obeys traffic rules: there are n arrows, parallel to the coordinate axes. The arrows do not intersect and do not touch each other. When the Beaveractor reaches some arrow, it turns in the arrow's direction and moves on until it either reaches the next arrow or gets outside the campus. The Beaveractor covers exactly one unit of space per one unit of time. You can assume that there are no obstacles to the Beaveractor.

The BSA scientists want to transport the brand new Beaveractor to the "Academic Tractor" research institute and send the Smart Beaver to do his postgraduate studies and sharpen pencils. They have q plans, representing the Beaveractor's initial position (xi, yi), the initial motion vector wi and the time ti that have passed after the escape started.

Your task is for each of the q plans to determine the Smart Beaver's position after the given time.

Input

The first line contains two integers: the number of traffic rules n and the size of the campus b, 0 ≤ n, 1 ≤ b. Next n lines contain the rules. Each line of the rules contains four space-separated integers x0, y0, x1, y1 — the beginning and the end of the arrow. It is guaranteed that all arrows are parallel to the coordinate axes and have no common points. All arrows are located inside the campus, that is, 0 ≤ x0, y0, x1, y1 ≤ b holds.

Next line contains integer q — the number of plans the scientists have, 1 ≤ q ≤ 105. The i-th plan is represented by two integers, xi, yi are the Beaveractor's coordinates at the initial time, 0 ≤ xi, yi ≤ b, character wi, that takes value U, D, L, R and sets the initial direction up, down, to the left or to the right correspondingly (the Y axis is directed upwards), and ti — the time passed after the escape started, 0 ≤ ti ≤ 1015.

  * to get 30 points you need to solve the problem with constraints n, b ≤ 30 (subproblem D1);
  * to get 60 points you need to solve the problem with constraints n, b ≤ 1000 (subproblems D1+D2);
  * to get 100 points you need to solve the problem with constraints n, b ≤ 105 (subproblems D1+D2+D3).

Output

Print q lines. Each line should contain two integers — the Beaveractor's coordinates at the final moment of time for each plan. If the Smart Beaver manages to leave the campus in time ti, print the coordinates of the last point in the campus he visited.

Examples

Input

3 3
0 0 0 1
0 2 2 2
3 3 2 3
12
0 0 L 0
0 0 L 1
0 0 L 2
0 0 L 3
0 0 L 4
0 0 L 5
0 0 L 6
2 0 U 2
2 0 U 3
3 0 U 5
1 3 D 2
1 3 R 2

Output

0 0
0 1
0 2
1 2
2 2
3 2
3 2
2 2
3 2
1 3
2 2
1 3


Given an n × n table T consisting of lowercase English letters. We'll consider some string s good if the table contains a correct path corresponding to the given string. In other words, good strings are all strings we can obtain by moving from the left upper cell of the table only to the right and down. Here's the formal definition of correct paths:

Consider rows of the table are numbered from 1 to n from top to bottom, and columns of the table are numbered from 1 to n from left to the right. Cell (r, c) is a cell of table T on the r-th row and in the c-th column. This cell corresponds to letter Tr, c.

A path of length k is a sequence of table cells [(r1, c1), (r2, c2), ..., (rk, ck)]. The following paths are correct:

  1. There is only one correct path of length 1, that is, consisting of a single cell: [(1, 1)];
  2. Let's assume that [(r1, c1), ..., (rm, cm)] is a correct path of length m, then paths [(r1, c1), ..., (rm, cm), (rm + 1, cm)] and [(r1, c1), ..., (rm, cm), (rm, cm + 1)] are correct paths of length m + 1.

We should assume that a path [(r1, c1), (r2, c2), ..., (rk, ck)] corresponds to a string of length k: Tr1, c1 + Tr2, c2 + ... + Trk, ck.

Two players play the following game: initially they have an empty string. Then the players take turns to add a letter to the end of the string. After each move (adding a new letter) the resulting string must be good. The game ends after 2n - 1 turns. A player wins by the following scenario:

  1. If the resulting string has strictly more letters "a" than letters "b", then the first player wins;
  2. If the resulting string has strictly more letters "b" than letters "a", then the second player wins;
  3. If the resulting string has the same number of letters "a" and "b", then the players end the game with a draw.

Your task is to determine the result of the game provided that both players played optimally well.

Input

The first line contains a single number n (1 ≤ n ≤ 20).

Next n lines contain n lowercase English letters each — table T.

Output

In a single line print string "FIRST", if the first player wins, "SECOND", if the second player wins and "DRAW", if the game ends with a draw.

Examples

Input

2
ab
cd

Output

DRAW

Input

2
xa
ay

Output

FIRST

Input

3
aab
bcb
bac

Output

DRAW

Note

Consider the first sample:

Good strings are strings: a, ab, ac, abd, acd.

The first player moves first and adds letter a to the string, as there is only one good string of length 1. Then the second player can add b or c and the game will end with strings abd or acd, correspondingly. In the first case it will be a draw (the string has one a and one b), in the second case the first player wins. Naturally, in this case the second player prefers to choose letter b and end the game with a draw.

Consider the second sample:

Good strings are: x, xa, xay.

We can see that the game will end with string xay and the first player wins.


Pavel loves grid mazes. A grid maze is an n × m rectangle maze where each cell is either empty, or is a wall. You can go from one cell to another only if both cells are empty and have a common side.

Pavel drew a grid maze with all empty cells forming a connected area. That is, you can go from any empty cell to any other one. Pavel doesn't like it when his maze has too little walls. He wants to turn exactly k empty cells into walls so that all the remaining cells still formed a connected area. Help him.

Input

The first line contains three integers n, m, k (1 ≤ n, m ≤ 500, 0 ≤ k < s), where n and m are the maze's height and width, correspondingly, k is the number of walls Pavel wants to add and letter s represents the number of empty cells in the original maze.

Each of the next n lines contains m characters. They describe the original maze. If a character on a line equals ".", then the corresponding cell is empty and if the character equals "#", then the cell is a wall.

Output

Print n lines containing m characters each: the new maze that fits Pavel's requirements. Mark the empty cells that you transformed into walls as "X", the other cells must be left without changes (that is, "." and "#").

It is guaranteed that a solution exists. If there are multiple solutions you can output any of them.

Examples

Input

3 4 2
#..#
..#.
#...

Output

#.X#
X.#.
#...

Input

5 4 5
#...
#.#.
.#..
...#
.#.#

Output

#XXX
#X#.
X#..
...#
.#.#


You all know the Dirichlet principle, the point of which is that if n boxes have no less than n + 1 items, that leads to the existence of a box in which there are at least two items.

Having heard of that principle, but having not mastered the technique of logical thinking, 8 year olds Stas and Masha invented a game. There are a different boxes and b different items, and each turn a player can either add a new box or a new item. The player, after whose turn the number of ways of putting b items into a boxes becomes no less then a certain given number n, loses. All the boxes and items are considered to be different. Boxes may remain empty.

Who loses if both players play optimally and Stas's turn is first?

Input

The only input line has three integers a, b, n (1 ≤ a ≤ 10000, 1 ≤ b ≤ 30, 2 ≤ n ≤ 109) — the initial number of the boxes, the number of the items and the number which constrains the number of ways, respectively. Guaranteed that the initial number of ways is strictly less than n.

Output

Output "Stas" if Masha wins. Output "Masha" if Stas wins. In case of a draw, output "Missing".

Examples

Input

2 2 10

Output

Masha

Input

5 5 16808

Output

Masha

Input

3 1 4

Output

Stas

Input

1 4 10

Output

Missing

Note

In the second example the initial number of ways is equal to 3125.

  * If Stas increases the number of boxes, he will lose, as Masha may increase the number of boxes once more during her turn. After that any Stas's move will lead to defeat.
  * But if Stas increases the number of items, then any Masha's move will be losing.


As usual, Sereja has array a, its elements are integers: a[1], a[2], ..., a[n]. Let's introduce notation:

<image>

A swap operation is the following sequence of actions:

  * choose two indexes i, j (i ≠ j);
  * perform assignments tmp = a[i], a[i] = a[j], a[j] = tmp.

What maximum value of function m(a) can Sereja get if he is allowed to perform at most k swap operations?

Input

The first line contains two integers n and k (1 ≤ n ≤ 200; 1 ≤ k ≤ 10). The next line contains n integers a[1], a[2], ..., a[n] ( - 1000 ≤ a[i] ≤ 1000).

Output

In a single line print the maximum value of m(a) that Sereja can get if he is allowed to perform at most k swap operations.

Examples

Input

10 2
10 -1 2 2 2 2 2 2 -1 10

Output

32

Input

5 10
-1 -1 -1 -1 -1

Output

-1


Jzzhu have n non-negative integers a1, a2, ..., an. We will call a sequence of indexes i1, i2, ..., ik (1 ≤ i1 < i2 < ... < ik ≤ n) a group of size k.

Jzzhu wonders, how many groups exists such that ai1 & ai2 & ... & aik = 0 (1 ≤ k ≤ n)? Help him and print this number modulo 1000000007 (109 + 7). Operation x & y denotes bitwise AND operation of two numbers.

Input

The first line contains a single integer n (1 ≤ n ≤ 106). The second line contains n integers a1, a2, ..., an (0 ≤ ai ≤ 106).

Output

Output a single integer representing the number of required groups modulo 1000000007 (109 + 7).

Examples

Input

3
2 3 3

Output

0

Input

4
0 1 2 3

Output

10

Input

6
5 2 0 5 2 1

Output

53


You are given a simple arithmetic expression of the form a?b, where a and b are integer constants, and ? can be one of the following operations: '+' (addition), '-' (subtraction), '*' (multiplication), '/' (integer division) or '%' (modulo operation).

Output the result of evaluation of this expression.

Input

The input is a single line containing an expression a?b. Here a and b are integers between 1 and 999, inclusive; ? is an operation character: '+', '-' (ASCII code 45), '*', '/' or '%'.

Output

Output a single integer — the result of evaluation of this expression.

Examples

Input

123+456

Output

579

Input

192/5

Output

38

Input

945%19

Output

14


Malek has recently found a treasure map. While he was looking for a treasure he found a locked door. There was a string s written on the door consisting of characters '(', ')' and '#'. Below there was a manual on how to open the door. After spending a long time Malek managed to decode the manual and found out that the goal is to replace each '#' with one or more ')' characters so that the final string becomes beautiful.

Below there was also written that a string is called beautiful if for each i (1 ≤ i ≤ |s|) there are no more ')' characters than '(' characters among the first i characters of s and also the total number of '(' characters is equal to the total number of ')' characters.

Help Malek open the door by telling him for each '#' character how many ')' characters he must replace it with.

Input

The first line of the input contains a string s (1 ≤ |s| ≤ 105). Each character of this string is one of the characters '(', ')' or '#'. It is guaranteed that s contains at least one '#' character.

Output

If there is no way of replacing '#' characters which leads to a beautiful string print  - 1. Otherwise for each character '#' print a separate line containing a positive integer, the number of ')' characters this character must be replaced with.

If there are several possible answers, you may output any of them.

Examples

Input

(((#)((#)

Output

1
2

Input

()((#((#(#()

Output

2
2
1

Input

#

Output

-1

Input

(#)

Output

-1

Note

|s| denotes the length of the string s.


After bracket sequences Arthur took up number theory. He has got a new favorite sequence of length n (a1, a2, ..., an), consisting of integers and integer k, not exceeding n.

This sequence had the following property: if you write out the sums of all its segments consisting of k consecutive elements (a1 + a2 ... + ak, a2 + a3 + ... + ak + 1, ..., an - k + 1 + an - k + 2 + ... + an), then those numbers will form strictly increasing sequence.

For example, for the following sample: n = 5, k = 3, a = (1, 2, 4, 5, 6) the sequence of numbers will look as follows: (1 + 2 + 4, 2 + 4 + 5, 4 + 5 + 6) = (7, 11, 15), that means that sequence a meets the described property.

Obviously the sequence of sums will have n - k + 1 elements.

Somebody (we won't say who) replaced some numbers in Arthur's sequence by question marks (if this number is replaced, it is replaced by exactly one question mark). We need to restore the sequence so that it meets the required property and also minimize the sum |ai|, where |ai| is the absolute value of ai.

Input

The first line contains two integers n and k (1 ≤ k ≤ n ≤ 105), showing how many numbers are in Arthur's sequence and the lengths of segments respectively.

The next line contains n space-separated elements ai (1 ≤ i ≤ n).

If ai = ?, then the i-th element of Arthur's sequence was replaced by a question mark.

Otherwise, ai ( - 109 ≤ ai ≤ 109) is the i-th element of Arthur's sequence.

Output

If Arthur is wrong at some point and there is no sequence that could fit the given information, print a single string "Incorrect sequence" (without the quotes).

Otherwise, print n integers — Arthur's favorite sequence. If there are multiple such sequences, print the sequence with the minimum sum |ai|, where |ai| is the absolute value of ai. If there are still several such sequences, you are allowed to print any of them. Print the elements of the sequence without leading zeroes.

Examples

Input

3 2
? 1 2

Output

0 1 2

Input

5 1
-10 -9 ? -7 -6

Output

-10 -9 -8 -7 -6

Input

5 3
4 6 7 2 9

Output

Incorrect sequence


You have multiset of n strings of the same length, consisting of lowercase English letters. We will say that those strings are easy to remember if for each string there is some position i and some letter c of the English alphabet, such that this string is the only string in the multiset that has letter c in position i.

For example, a multiset of strings {"abc", "aba", "adc", "ada"} are not easy to remember. And multiset {"abc", "ada", "ssa"} is easy to remember because:

  * the first string is the only string that has character c in position 3;
  * the second string is the only string that has character d in position 2;
  * the third string is the only string that has character s in position 2.

You want to change your multiset a little so that it is easy to remember. For aij coins, you can change character in the j-th position of the i-th string into any other lowercase letter of the English alphabet. Find what is the minimum sum you should pay in order to make the multiset of strings easy to remember.

Input

The first line contains two integers n, m (1 ≤ n, m ≤ 20) — the number of strings in the multiset and the length of the strings respectively. Next n lines contain the strings of the multiset, consisting only of lowercase English letters, each string's length is m.

Next n lines contain m integers each, the i-th of them contains integers ai1, ai2, ..., aim (0 ≤ aij ≤ 106).

Output

Print a single number — the answer to the problem.

Examples

Input

4 5
abcde
abcde
abcde
abcde
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1
1 1 1 1 1

Output

3

Input

4 3
abc
aba
adc
ada
10 10 10
10 1 10
10 10 10
10 1 10

Output

2

Input

3 3
abc
ada
ssa
1 1 1
1 1 1
1 1 1

Output

0


You've got array A, consisting of n integers and a positive integer k. Array A is indexed by integers from 1 to n.

You need to permute the array elements so that value

<image> became minimal possible. In particular, it is allowed not to change order of elements at all.

Input

The first line contains two integers n, k (2 ≤ n ≤ 3·105, 1 ≤ k ≤ min(5000, n - 1)).

The second line contains n integers A[1], A[2], ..., A[n] ( - 109 ≤ A[i] ≤ 109), separate by spaces — elements of the array A.

Output

Print the minimum possible value of the sum described in the statement.

Examples

Input

3 2
1 2 4

Output

1

Input

5 2
3 -5 3 -5 3

Output

0

Input

6 3
4 3 4 3 2 5

Output

3

Note

In the first test one of the optimal permutations is 1 4 2.

In the second test the initial order is optimal.

In the third test one of the optimal permutations is 2 3 4 4 3 5.


BCPC stands for Byteforces Collegiate Programming Contest, and is the most famous competition in Byteforces.

BCPC is a team competition. Each team is composed by a coach and three contestants. Blenda is the coach of the Bit State University(BSU), and she is very strict selecting the members of her team.

<image>

In BSU there are n students numbered from 1 to n. Since all BSU students are infinitely smart, the only important parameters for Blenda are their reading and writing speed. After a careful measuring, Blenda have found that the i-th student have a reading speed equal to ri (words per minute), and a writing speed of wi (symbols per minute). Since BSU students are very smart, the measured speeds are sometimes very big and Blenda have decided to subtract some constant value c from all the values of reading speed and some value d from all the values of writing speed. Therefore she considers ri' = ri - c and wi' = wi - d.

The student i is said to overwhelm the student j if and only if ri'·wj' > rj'·wi'. Blenda doesn’t like fights in teams, so she thinks that a team consisting of three distinct students i, j and k is good if i overwhelms j, j overwhelms k, and k overwhelms i. Yes, the relation of overwhelming is not transitive as it often happens in real life.

Since Blenda is busy preparing a training camp in Codeforces, you are given a task to calculate the number of different good teams in BSU. Two teams are considered to be different if there is at least one student that is present in one team but is not present in the other. In other words, two teams are different if the sets of students that form these teams are different.

Input

In the first line of the input three integers n, c and d (3 ≤ n ≤ 345678, 1 ≤ c, d ≤ 109) are written. They denote the number of students Blenda can use to form teams, the value subtracted from all reading speeds and the value subtracted from all writing speeds respectively.

Each of the next n lines contains two integers ri and wi (0 < ri, wi ≤ 109, |ri - c| + |wi - d| > 0). There are no two students, such that both their reading and writing speeds coincide, i.e. for every i ≠ j condition |ri - rj| + |wi - wj| > 0 holds.

Output

Print the number of different teams in BSU, that are good according to Blenda's definition.

Examples

Input

5 2 2
1 1
4 1
2 3
3 2
3 4

Output

4

Input

7 6 6
3 2
1 7
5 7
3 7
6 4
8 9
8 5

Output

11

Note

In the first sample the following teams are good: (i = 1, j = 2, k = 3), (i = 2, j = 5, k = 1), (i = 1, j = 4, k = 3), (i = 5, j = 1, k = 4).

Note, that for example the team (i = 3, j = 1, k = 2) is also good, but is considered to be the same as the team (i = 1, j = 2, k = 3).


Peter got a new snow blower as a New Year present. Of course, Peter decided to try it immediately. After reading the instructions he realized that it does not work like regular snow blowing machines. In order to make it work, you need to tie it to some point that it does not cover, and then switch it on. As a result it will go along a circle around this point and will remove all the snow from its path.

Formally, we assume that Peter's machine is a polygon on a plane. Then, after the machine is switched on, it will make a circle around the point to which Peter tied it (this point lies strictly outside the polygon). That is, each of the points lying within or on the border of the polygon will move along the circular trajectory, with the center of the circle at the point to which Peter tied his machine.

Peter decided to tie his car to point P and now he is wondering what is the area of the region that will be cleared from snow. Help him.

Input

The first line of the input contains three integers — the number of vertices of the polygon n (<image>), and coordinates of point P.

Each of the next n lines contains two integers — coordinates of the vertices of the polygon in the clockwise or counterclockwise order. It is guaranteed that no three consecutive vertices lie on a common straight line.

All the numbers in the input are integers that do not exceed 1 000 000 in their absolute value.

Output

Print a single real value number — the area of the region that will be cleared. Your answer will be considered correct if its absolute or relative error does not exceed 10 - 6.

Namely: let's assume that your answer is a, and the answer of the jury is b. The checker program will consider your answer correct, if <image>.

Examples

Input

3 0 0
0 1
-1 2
1 2

Output

12.566370614359172464

Input

4 1 -1
0 0
1 2
2 0
1 1

Output

21.991148575128551812

Note

In the first sample snow will be removed from that area:

<image>


Yash is finally tired of computing the length of the longest Fibonacci-ish sequence. He now plays around with more complex things such as Fibonacci-ish potentials.

Fibonacci-ish potential of an array ai is computed as follows:

  1. Remove all elements j if there exists i < j such that ai = aj.
  2. Sort the remaining elements in ascending order, i.e. a1 < a2 < ... < an.
  3. Compute the potential as P(a) = a1·F1 + a2·F2 + ... + an·Fn, where Fi is the i-th Fibonacci number (see notes for clarification).

You are given an array ai of length n and q ranges from lj to rj. For each range j you have to compute the Fibonacci-ish potential of the array bi, composed using all elements of ai from lj to rj inclusive. Find these potentials modulo m.

Input

The first line of the input contains integers of n and m (1 ≤ n, m ≤ 30 000) — the length of the initial array and the modulo, respectively.

The next line contains n integers ai (0 ≤ ai ≤ 109) — elements of the array.

Then follow the number of ranges q (1 ≤ q ≤ 30 000).

Last q lines contain pairs of indices li and ri (1 ≤ li ≤ ri ≤ n) — ranges to compute Fibonacci-ish potentials.

Output

Print q lines, i-th of them must contain the Fibonacci-ish potential of the i-th range modulo m.

Example

Input

5 10
2 1 2 1 2
2
2 4
4 5

Output

3
3

Note

For the purpose of this problem define Fibonacci numbers as follows:

  1. F1 = F2 = 1.
  2. Fn = Fn - 1 + Fn - 2 for each n > 2.

In the first query, the subarray [1,2,1] can be formed using the minimal set {1,2}. Thus, the potential of this subarray is 1*1+2*1=3.


You are given a table consisting of n rows and m columns. Each cell of the table contains either 0 or 1. In one move, you are allowed to pick any row or any column and invert all values, that is, replace 0 by 1 and vice versa.

What is the minimum number of cells with value 1 you can get after applying some number of operations?

Input

The first line of the input contains two integers n and m (1 ≤ n ≤ 20, 1 ≤ m ≤ 100 000) — the number of rows and the number of columns, respectively.

Then n lines follows with the descriptions of the rows. Each line has length m and contains only digits '0' and '1'.

Output

Output a single integer — the minimum possible number of ones you can get after applying some sequence of operations.

Example

Input

3 4
0110
1010
0111

Output

2


Recently, Pari and Arya did some research about NP-Hard problems and they found the minimum vertex cover problem very interesting.

Suppose the graph G is given. Subset A of its vertices is called a vertex cover of this graph, if for each edge uv there is at least one endpoint of it in this set, i.e. <image> or <image> (or both).

Pari and Arya have won a great undirected graph as an award in a team contest. Now they have to split it in two parts, but both of them want their parts of the graph to be a vertex cover.

They have agreed to give you their graph and you need to find two disjoint subsets of its vertices A and B, such that both A and B are vertex cover or claim it's impossible. Each vertex should be given to no more than one of the friends (or you can even keep it for yourself).

Input

The first line of the input contains two integers n and m (2 ≤ n ≤ 100 000, 1 ≤ m ≤ 100 000) — the number of vertices and the number of edges in the prize graph, respectively.

Each of the next m lines contains a pair of integers ui and vi (1 ≤ ui, vi ≤ n), denoting an undirected edge between ui and vi. It's guaranteed the graph won't contain any self-loops or multiple edges.

Output

If it's impossible to split the graph between Pari and Arya as they expect, print "-1" (without quotes).

If there are two disjoint sets of vertices, such that both sets are vertex cover, print their descriptions. Each description must contain two lines. The first line contains a single integer k denoting the number of vertices in that vertex cover, and the second line contains k integers — the indices of vertices. Note that because of m ≥ 1, vertex cover cannot be empty.

Examples

Input

4 2
1 2
2 3

Output

1
2
2
1 3

Input

3 3
1 2
2 3
1 3

Output

-1

Note

In the first sample, you can give the vertex number 2 to Arya and vertices numbered 1 and 3 to Pari and keep vertex number 4 for yourself (or give it someone, if you wish).

In the second sample, there is no way to satisfy both Pari and Arya.


Tree is a connected acyclic graph. Suppose you are given a tree consisting of n vertices. The vertex of this tree is called centroid if the size of each connected component that appears if this vertex is removed from the tree doesn't exceed <image>.

You are given a tree of size n and can perform no more than one edge replacement. Edge replacement is the operation of removing one edge from the tree (without deleting incident vertices) and inserting one new edge (without adding new vertices) in such a way that the graph remains a tree. For each vertex you have to determine if it's possible to make it centroid by performing no more than one edge replacement.

Input

The first line of the input contains an integer n (2 ≤ n ≤ 400 000) — the number of vertices in the tree. Each of the next n - 1 lines contains a pair of vertex indices ui and vi (1 ≤ ui, vi ≤ n) — endpoints of the corresponding edge.

Output

Print n integers. The i-th of them should be equal to 1 if the i-th vertex can be made centroid by replacing no more than one edge, and should be equal to 0 otherwise.

Examples

Input

3
1 2
2 3

Output

1 1 1

Input

5
1 2
1 3
1 4
1 5

Output

1 0 0 0 0

Note

In the first sample each vertex can be made a centroid. For example, in order to turn vertex 1 to centroid one have to replace the edge (2, 3) with the edge (1, 3).


Polycarp starts his own business. Tomorrow will be the first working day of his car repair shop. For now the car repair shop is very small and only one car can be repaired at a given time.

Polycarp is good at marketing, so he has already collected n requests from clients. The requests are numbered from 1 to n in order they came.

The i-th request is characterized by two values: si — the day when a client wants to start the repair of his car, di — duration (in days) to repair the car. The days are enumerated from 1, the first day is tomorrow, the second day is the day after tomorrow and so on.

Polycarp is making schedule by processing requests in the order from the first to the n-th request. He schedules the i-th request as follows:

  * If the car repair shop is idle for di days starting from si (si, si + 1, ..., si + di - 1), then these days are used to repair a car of the i-th client.
  * Otherwise, Polycarp finds the first day x (from 1 and further) that there are di subsequent days when no repair is scheduled starting from x. In other words he chooses the smallest positive x that all days x, x + 1, ..., x + di - 1 are not scheduled for repair of any car. So, the car of the i-th client will be repaired in the range [x, x + di - 1]. It is possible that the day x when repair is scheduled to start will be less than si.

Given n requests, you are asked to help Polycarp schedule all of them according to the rules above.

Input

The first line contains integer n (1 ≤ n ≤ 200) — the number of requests from clients.

The following n lines contain requests, one request per line. The i-th request is given as the pair of integers si, di (1 ≤ si ≤ 109, 1 ≤ di ≤ 5·106), where si is the preferred time to start repairing the i-th car, di is the number of days to repair the i-th car.

The requests should be processed in the order they are given in the input.

Output

Print n lines. The i-th line should contain two integers — the start day to repair the i-th car and the finish day to repair the i-th car.

Examples

Input

3
9 2
7 3
2 4

Output

9 10
1 3
4 7

Input

4
1000000000 1000000
1000000000 1000000
100000000 1000000
1000000000 1000000

Output

1000000000 1000999999
1 1000000
100000000 100999999
1000001 2000000


Ilya is an experienced player in tic-tac-toe on the 4 × 4 field. He always starts and plays with Xs. He played a lot of games today with his friend Arseny. The friends became tired and didn't finish the last game. It was Ilya's turn in the game when they left it. Determine whether Ilya could have won the game by making single turn or not.

The rules of tic-tac-toe on the 4 × 4 field are as follows. Before the first turn all the field cells are empty. The two players take turns placing their signs into empty cells (the first player places Xs, the second player places Os). The player who places Xs goes first, the another one goes second. The winner is the player who first gets three of his signs in a row next to each other (horizontal, vertical or diagonal).

Input

The tic-tac-toe position is given in four lines.

Each of these lines contains four characters. Each character is '.' (empty cell), 'x' (lowercase English letter x), or 'o' (lowercase English letter o). It is guaranteed that the position is reachable playing tic-tac-toe, and it is Ilya's turn now (in particular, it means that the game is not finished). It is possible that all the cells are empty, it means that the friends left without making single turn.

Output

Print single line: "YES" in case Ilya could have won by making single turn, and "NO" otherwise.

Examples

Input

xx..
.oo.
x...
oox.

Output

YES

Input

x.ox
ox..
x.o.
oo.x

Output

NO

Input

x..x
..oo
o...
x.xo

Output

YES

Input

o.x.
o...
.x..
ooxx

Output

NO

Note

In the first example Ilya had two winning moves: to the empty cell in the left column and to the leftmost empty cell in the first row.

In the second example it wasn't possible to win by making single turn.

In the third example Ilya could have won by placing X in the last row between two existing Xs.

In the fourth example it wasn't possible to win by making single turn.


Stepan likes to repeat vowel letters when he writes words. For example, instead of the word "pobeda" he can write "pobeeeedaaaaa".

Sergey does not like such behavior, so he wants to write a program to format the words written by Stepan. This program must combine all consecutive equal vowels to a single vowel. The vowel letters are "a", "e", "i", "o", "u" and "y".

There are exceptions: if letters "e" or "o" repeat in a row exactly 2 times, like in words "feet" and "foot", the program must skip them and do not transform in one vowel. For example, the word "iiiimpleeemeentatiioon" must be converted to the word "implemeentatioon".

Sergey is very busy and asks you to help him and write the required program.

Input

The first line contains the integer n (1 ≤ n ≤ 100 000) — the number of letters in the word written by Stepan.

The second line contains the string s which has length that equals to n and contains only lowercase English letters — the word written by Stepan.

Output

Print the single string — the word written by Stepan converted according to the rules described in the statement.

Examples

Input

13
pobeeeedaaaaa

Output

pobeda

Input

22
iiiimpleeemeentatiioon

Output

implemeentatioon

Input

18
aeiouyaaeeiioouuyy

Output

aeiouyaeeioouy

Input

24
aaaoooiiiuuuyyyeeeggghhh

Output

aoiuyeggghhh


Finally Fox Ciel arrived in front of her castle!

She have to type a password to enter her castle. An input device attached to her castle is a bit unusual.

The input device is a 1 × n rectangle divided into n square panels. They are numbered 1 to n from left to right. Each panel has a state either ON or OFF. Initially all panels are in the OFF state. She can enter her castle if and only if x1-th, x2-th, ..., xk-th panels are in the ON state and other panels are in the OFF state.

She is given an array a1, ..., al. In each move, she can perform the following operation: choose an index i (1 ≤ i ≤ l), choose consecutive ai panels, and flip the states of those panels (i.e. ON → OFF, OFF → ON).

Unfortunately she forgets how to type the password with only above operations. Determine the minimal number of operations required to enter her castle.

Input

The first line contains three integers n, k and l (1 ≤ n ≤ 10000, 1 ≤ k ≤ 10, 1 ≤ l ≤ 100), separated by single spaces.

The second line contains k integers x1, ..., xk (1 ≤ x1 < x2 < ... < xk ≤ n), separated by single spaces.

The third line contains l integers a1, ..., al (1 ≤ ai ≤ n), separated by single spaces. It is possible that some elements of the array ai are equal value.

Output

Print the minimal number of moves required to type the password. If it's impossible, print -1.

Examples

Input

10 8 2
1 2 3 5 6 7 8 9
3 5

Output

2

Input

3 2 1
1 2
3

Output

-1

Note

One possible way to type the password in the first example is following: In the first move, choose 1st, 2nd, 3rd panels and flip those panels. In the second move, choose 5th, 6th, 7th, 8th, 9th panels and flip those panels.


Mister B once received a gift: it was a book about aliens, which he started read immediately. This book had c pages.

At first day Mister B read v0 pages, but after that he started to speed up. Every day, starting from the second, he read a pages more than on the previous day (at first day he read v0 pages, at second — v0 + a pages, at third — v0 + 2a pages, and so on). But Mister B is just a human, so he physically wasn't able to read more than v1 pages per day.

Also, to refresh his memory, every day, starting from the second, Mister B had to reread last l pages he read on the previous day. Mister B finished the book when he read the last page for the first time.

Help Mister B to calculate how many days he needed to finish the book.

Input

First and only line contains five space-separated integers: c, v0, v1, a and l (1 ≤ c ≤ 1000, 0 ≤ l < v0 ≤ v1 ≤ 1000, 0 ≤ a ≤ 1000) — the length of the book in pages, the initial reading speed, the maximum reading speed, the acceleration in reading speed and the number of pages for rereading.

Output

Print one integer — the number of days Mister B needed to finish the book.

Examples

Input

5 5 10 5 4

Output

1

Input

12 4 12 4 1

Output

3

Input

15 1 100 0 0

Output

15

Note

In the first sample test the book contains 5 pages, so Mister B read it right at the first day.

In the second sample test at first day Mister B read pages number 1 - 4, at second day — 4 - 11, at third day — 11 - 12 and finished the book.

In third sample test every day Mister B read 1 page of the book, so he finished in 15 days.


Polycarp takes part in a math show. He is given n tasks, each consists of k subtasks, numbered 1 through k. It takes him tj minutes to solve the j-th subtask of any task. Thus, time required to solve a subtask depends only on its index, but not on the task itself. Polycarp can solve subtasks in any order.

By solving subtask of arbitrary problem he earns one point. Thus, the number of points for task is equal to the number of solved subtasks in it. Moreover, if Polycarp completely solves the task (solves all k of its subtasks), he recieves one extra point. Thus, total number of points he recieves for the complete solution of the task is k + 1.

Polycarp has M minutes of time. What is the maximum number of points he can earn?

Input

The first line contains three integer numbers n, k and M (1 ≤ n ≤ 45, 1 ≤ k ≤ 45, 0 ≤ M ≤ 2·109).

The second line contains k integer numbers, values tj (1 ≤ tj ≤ 1000000), where tj is the time in minutes required to solve j-th subtask of any task.

Output

Print the maximum amount of points Polycarp can earn in M minutes.

Examples

Input

3 4 11
1 2 3 4

Output

6

Input

5 5 10
1 2 4 8 16

Output

7

Note

In the first example Polycarp can complete the first task and spend 1 + 2 + 3 + 4 = 10 minutes. He also has the time to solve one subtask of the second task in one minute.

In the second example Polycarp can solve the first subtask of all five tasks and spend 5·1 = 5 minutes. Also he can solve the second subtasks of two tasks and spend 2·2 = 4 minutes. Thus, he earns 5 + 2 = 7 points in total.


You're trying to set the record on your favorite video game. The game consists of N levels, which must be completed sequentially in order to beat the game. You usually complete each level as fast as possible, but sometimes finish a level slower. Specifically, you will complete the i-th level in either Fi seconds or Si seconds, where Fi < Si, and there's a Pi percent chance of completing it in Fi seconds. After completing a level, you may decide to either continue the game and play the next level, or reset the game and start again from the first level. Both the decision and the action are instant.

Your goal is to complete all the levels sequentially in at most R total seconds. You want to minimize the expected amount of time playing before achieving that goal. If you continue and reset optimally, how much total time can you expect to spend playing?

Input

The first line of input contains integers N and R <image>, the number of levels and number of seconds you want to complete the game in, respectively. N lines follow. The ith such line contains integers Fi, Si, Pi (1 ≤ Fi < Si ≤ 100, 80 ≤ Pi ≤ 99), the fast time for level i, the slow time for level i, and the probability (as a percentage) of completing level i with the fast time.

Output

Print the total expected time. Your answer must be correct within an absolute or relative error of 10 - 9.

Formally, let your answer be a, and the jury's answer be b. Your answer will be considered correct, if <image>.

Examples

Input

1 8
2 8 81

Output

3.14

Input

2 30
20 30 80
3 9 85

Output

31.4

Input

4 319
63 79 89
79 97 91
75 87 88
75 90 83

Output

314.159265358

Note

In the first example, you never need to reset. There's an 81% chance of completing the level in 2 seconds and a 19% chance of needing 8 seconds, both of which are within the goal time. The expected time is 0.81·2 + 0.19·8 = 3.14.

In the second example, you should reset after the first level if you complete it slowly. On average it will take 0.25 slow attempts before your first fast attempt. Then it doesn't matter whether you complete the second level fast or slow. The expected time is 0.25·30 + 20 + 0.85·3 + 0.15·9 = 31.4.


Alex, Bob and Carl will soon participate in a team chess tournament. Since they are all in the same team, they have decided to practise really hard before the tournament. But it's a bit difficult for them because chess is a game for two players, not three.

So they play with each other according to following rules:

  * Alex and Bob play the first game, and Carl is spectating;
  * When the game ends, the one who lost the game becomes the spectator in the next game, and the one who was spectating plays against the winner.

Alex, Bob and Carl play in such a way that there are no draws.

Today they have played n games, and for each of these games they remember who was the winner. They decided to make up a log of games describing who won each game. But now they doubt if the information in the log is correct, and they want to know if the situation described in the log they made up was possible (that is, no game is won by someone who is spectating if Alex, Bob and Carl play according to the rules). Help them to check it!

Input

The first line contains one integer n (1 ≤ n ≤ 100) — the number of games Alex, Bob and Carl played.

Then n lines follow, describing the game log. i-th line contains one integer ai (1 ≤ ai ≤ 3) which is equal to 1 if Alex won i-th game, to 2 if Bob won i-th game and 3 if Carl won i-th game.

Output

Print YES if the situation described in the log was possible. Otherwise print NO.

Examples

Input

3
1
1
2

Output

YES

Input

2
1
2

Output

NO

Note

In the first example the possible situation is:

  1. Alex wins, Carl starts playing instead of Bob;
  2. Alex wins, Bob replaces Carl;
  3. Bob wins.

The situation in the second example is impossible because Bob loses the first game, so he cannot win the second one.


Given a string s, process q queries, each having one of the following forms:

  * 1 i c — Change the i-th character in the string to c.
  * 2 l r y — Consider the substring of s starting at position l and ending at position r. Output the number of times y occurs as a substring in it.

Input

The first line of the input contains the string s (1 ≤ |s| ≤ 105) of lowercase English letters.

The second line contains an integer q (1 ≤ q ≤ 105) — the number of queries to process.

The next q lines describe the queries and may have one of the following forms:

  * 1 i c (1 ≤ i ≤ |s|)
  * 2 l r y (1 ≤ l ≤ r ≤ |s|)

c is a lowercase English letter and y is a non-empty string consisting of only lowercase English letters.

The sum of |y| over all queries of second type is at most 105.

It is guaranteed that there is at least one query of second type.

All strings are 1-indexed.

|s| is the length of the string s.

Output

For each query of type 2, output the required answer in a separate line.

Examples

Input

ababababa
3
2 1 7 aba
1 5 c
2 1 7 aba

Output

3
1

Input

abcdcbc
5
2 1 7 bc
1 4 b
2 4 7 bc
1 2 a
2 1 4 aa

Output

2
2
1

Note

Consider the first sample case. Initially, the string aba occurs 3 times in the range [1, 7]. Note that two occurrences may overlap.

After the update, the string becomes ababcbaba and now aba occurs only once in the range [1, 7].


Petya and Vasya arranged a game. The game runs by the following rules. Players have a directed graph consisting of n vertices and m edges. One of the vertices contains a chip. Initially the chip is located at vertex s. Players take turns moving the chip along some edge of the graph. Petya goes first. Player who can't move the chip loses. If the game lasts for 106 turns the draw is announced.

Vasya was performing big laboratory work in "Spelling and parts of speech" at night before the game, so he fell asleep at the very beginning of the game. Petya decided to take the advantage of this situation and make both Petya's and Vasya's moves.

Your task is to help Petya find out if he can win the game or at least draw a tie.

Input

The first line of input contain two integers n and m — the number of vertices and the number of edges in the graph (2 ≤ n ≤ 105, 0 ≤ m ≤ 2·105).

The next n lines contain the information about edges of the graph. i-th line (1 ≤ i ≤ n) contains nonnegative integer ci — number of vertices such that there is an edge from i to these vertices and ci distinct integers ai, j — indices of these vertices (1 ≤ ai, j ≤ n, ai, j ≠ i).

It is guaranteed that the total sum of ci equals to m.

The next line contains index of vertex s — the initial position of the chip (1 ≤ s ≤ n).

Output

If Petya can win print «Win» in the first line. In the next line print numbers v1, v2, ..., vk (1 ≤ k ≤ 106) — the sequence of vertices Petya should visit for the winning. Vertex v1 should coincide with s. For i = 1... k - 1 there should be an edge from vi to vi + 1 in the graph. There must be no possible move from vertex vk. The sequence should be such that Petya wins the game.

If Petya can't win but can draw a tie, print «Draw» in the only line. Otherwise print «Lose».

Examples

Input

5 6
2 2 3
2 4 5
1 4
1 5
0
1

Output

Win
1 2 4 5

Input

3 2
1 3
1 1
0
2

Output

Lose

Input

2 2
1 2
1 1
1

Output

Draw

Note

In the first example the graph is the following:

<image>

Initially the chip is located at vertex 1. In the first move Petya moves the chip to vertex 2, after that he moves it to vertex 4 for Vasya. After that he moves to vertex 5. Now it is Vasya's turn and there is no possible move, so Petya wins.

In the second example the graph is the following:

<image>

Initially the chip is located at vertex 2. The only possible Petya's move is to go to vertex 1. After that he has to go to 3 for Vasya. Now it's Petya's turn but he has no possible move, so Petya loses.

In the third example the graph is the following:

<image>

Petya can't win, but he can move along the cycle, so the players will draw a tie.


You are given an undirected graph, consisting of n vertices and m edges. The graph does not necessarily connected. Guaranteed, that the graph does not contain multiple edges (more than one edges between a pair of vertices) or loops (edges from a vertex to itself).

A cycle in a graph is called a simple, if it contains each own vertex exactly once. So simple cycle doesn't allow to visit a vertex more than once in a cycle.

Determine the edges, which belong to exactly on one simple cycle.

Input

The first line contain two integers n and m (1 ≤ n ≤ 100 000, 0 ≤ m ≤ min(n ⋅ (n - 1) / 2, 100 000)) — the number of vertices and the number of edges.

Each of the following m lines contain two integers u and v (1 ≤ u, v ≤ n, u ≠ v) — the description of the edges.

Output

In the first line print the number of edges, which belong to exactly one simple cycle.

In the second line print the indices of edges, which belong to exactly one simple cycle, in increasing order. The edges are numbered from one in the same order as they are given in the input.

Examples

Input

3 3
1 2
2 3
3 1

Output

3
1 2 3

Input

6 7
2 3
3 4
4 2
1 2
1 5
5 6
6 1

Output

6
1 2 3 5 6 7

Input

5 6
1 2
2 3
2 4
4 3
2 5
5 3

Output

0


You have a Petri dish with bacteria and you are preparing to dive into the harsh micro-world. But, unfortunately, you don't have any microscope nearby, so you can't watch them.

You know that you have n bacteria in the Petri dish and size of the i-th bacteria is a_i. Also you know intergalactic positive integer constant K.

The i-th bacteria can swallow the j-th bacteria if and only if a_i > a_j and a_i ≤ a_j + K. The j-th bacteria disappear, but the i-th bacteria doesn't change its size. The bacteria can perform multiple swallows. On each swallow operation any bacteria i can swallow any bacteria j if a_i > a_j and a_i ≤ a_j + K. The swallow operations go one after another.

For example, the sequence of bacteria sizes a=[101, 53, 42, 102, 101, 55, 54] and K=1. The one of possible sequences of swallows is: [101, 53, 42, 102, \underline{101}, 55, 54] → [101, \underline{53}, 42, 102, 55, 54] → [\underline{101}, 42, 102, 55, 54] → [42, 102, 55, \underline{54}] → [42, 102, 55]. In total there are 3 bacteria remained in the Petri dish.

Since you don't have a microscope, you can only guess, what the minimal possible number of bacteria can remain in your Petri dish when you finally will find any microscope.

Input

The first line contains two space separated positive integers n and K (1 ≤ n ≤ 2 ⋅ 10^5, 1 ≤ K ≤ 10^6) — number of bacteria and intergalactic constant K.

The second line contains n space separated integers a_1, a_2, ..., a_n (1 ≤ a_i ≤ 10^6) — sizes of bacteria you have.

Output

Print the only integer — minimal possible number of bacteria can remain.

Examples

Input

7 1
101 53 42 102 101 55 54

Output

3

Input

6 5
20 15 10 15 20 25

Output

1

Input

7 1000000
1 1 1 1 1 1 1

Output

7

Note

The first example is clarified in the problem statement.

In the second example an optimal possible sequence of swallows is: [20, 15, 10, 15, \underline{20}, 25] → [20, 15, 10, \underline{15}, 25] → [20, 15, \underline{10}, 25] → [20, \underline{15}, 25] → [\underline{20}, 25] → [25].

In the third example no bacteria can swallow any other bacteria.


Coach Ankit is forming a team for the Annual Inter Galactic Relay Race. He has N students that train under him and he knows their strengths. The strength of a student is represented by a positive integer.

The coach has to form a team of K students. The strength of a team is defined by the strength of the weakest student in the team. Now he wants to know the sum of strengths of all the teams of size K that can be formed modulo 1000000007. Please help him.

Input
The first line contains the number of test cases T.
Each case begins with a line containing integers N and K. The next line contains N space-separated numbers which describe the strengths of the students.

Output
For test case output a single integer, the answer as described in the problem statement.

Constraints:
1 ≤ T ≤ 100
1 ≤ N ≤ 100000
1 ≤ K ≤ N
0 ≤ Strength of each student ≤ 2000000000
Strength of all the students are different.

SAMPLE INPUT
2
2 1
5 4
3 2
1 0 2

SAMPLE OUTPUT
9
1

Explanation

For first test case: 5+4=9, as team can only consist of 1 student.
For second test case: min(1,0) + min(1,2) + min(0,2) = 0+1+0 =1


Continuing from previous version of codeXplod series i.e. CodeXplod 1.0,Chandu and daspal are still fighting over a matter of MOMO's(they are very fond of Mo Mos of sector 3..:P).This time the fight became so savior that they want to kill each other.As we all  know that during a fight it is most probable outcome that both will be injured.As their friend we don`t want that to happen so we devise a method.

We gave them two integers (x,y) as they have to play a game on these values.
Game is defined as follows:-
Players play alternative.
1-During a move they are allowed to subtract gcd(x,y) from both x and y .

2-The player which makes either one of the values to zero or both to zero will win the game and will eat all the MOMO's.

3-If neither of values after 1st step does not equal to zero then the second player gets a chance and play as described by rule 1,2,3.
 Now,You are given two integers and you have to find out who will eat all the MOMOs.

 Input Format:-

 First Line of input will contain an integer T denoting number of test cases.
 Each test cases contains 2 integers(x,y) with a strings.
 Two integers  x,y specify the integers given to them by us and string will specify who will play first i.e. String will be either of values "Chandu" and "Daspal"(Quotes only for clarity).
 Output Format:-

 Output a string either "Chandu" or "Daspal"(Qoutes for clarity) determinging who will win the game.
 Constraints:-

1 ≤ T ≤ 50

1 ≤ x,y ≤ 3000

SAMPLE INPUT
1
2 3 Chandu

SAMPLE OUTPUT
Daspal

Explanation

First gcd(2,3)=1 so Chandu subtract 1 from each so new pair now (1,2) now its Daspal's Turn he has gcd(1,2)=1 so after subtracting (0,1) now daspal have one zero so he is declared as winner of the game.Phew!!!


Bosky is a very curious child who turned 13 yesterday. His parents gifted him a digital watch which he really liked. He was amazed to see how each number can be represented inside one cell only by switching different edges on and off.

Today, while he was in his home alone, getting bored looking at his watch, an idea popped up in his scientific mind. He immediately went to the garage and picked up a long rectangular cardboard, which was 2 units wide and L units long, and lots of thin LED tubes which were 1 unit long.

Bosky wanted to place LED tubes on the rectangular cardboard in such a way that he can represent as many digits as possible using the LED tubes and rectangular board similar to how digits are represented on his watch.

He realized that none of the LED tubes are working so he decided to purchase new LED tubes.

Each LED tube will cost Rs. 7

He has Rs. M with him and he now goes to market and spends all the money purchasing LED tubes.

After he brought all the LED tubes home, he wondered how many different digits he can represent on the rectangular board such that he doesn't have to repeat a number again.

But Bosky is tired now and does not want to try all the possible combinations to figure out how many digits he can represent on the board. He knows that you are a great programmer and you would surely help him. He calls you and tells you the length L and the amount of money M he has. Now, you have to tell him how many different digits he can represent on the rectangular board such that none of the digits is repeated, given:

0 requires -- 6 tubes

1 requires -- 2 tubes

2 requires -- 5 tubes

3 requires -- 5 tubes

4 requires -- 4 tubes

5 requires -- 5 tubes

6 requires -- 6 tubes

7 requires -- 3 tubes

8 requires -- 7 tubes

9 requires -- 6 tubes

[NOTE: Each digit occupies exactly 1 unit length on the board]

Input:

Input will contain a number T denoting the number of test cases.

Then T test cases follow, each one consisting of two space-separated integers L and M .

Output

For each test case, output a single integer - the number of different digits Bosky can represent on the rectangular board such that none of the number is repeated and all the digits fit on the board.

Constraints

1  ≤ T  ≤ 100

0 ≤ M ≤ 2000

0 ≤ L ≤ 10

SAMPLE INPUT
4
1 14
1 50
4 28
5 700

SAMPLE OUTPUT
1
10
3
32491

Explanation

Case 1: He has 14 rupees and he can purchase 2 LED tubes using which he can represent only '1' on the board.

Case 2: He can purchase 7 LED tubes using 50 rupees and the number of digits he can represent using 7 tubes are 10.


Professor just has checked all the N students tests.  Everything was fine but then he realised that none of the students had signed their papers, so he doesn't know which test belongs to which student.

But it's definitely not professors's job to catch every student and asked him to find his paper! So he will hand out these papers in a random way.

Now he is interested in the following question: what is the probability that X students  will receive someone other's test, not their where L ≤ X ≤ R.

Input:

The first line contains 3 space-separated integers: N, L, R.

Output:

Let's suppose the answer is a fraction P / Q where P and Q are coprime. Output P * Q^-1  modulo 10^9 + 7.

Constraints:

1 ≤ N ≤ 100
0 ≤ L ≤ R ≤ N

SAMPLE INPUT
3 1 3

SAMPLE OUTPUT
833333340

Explanation

It's not possible that exactly 1 students doesn't receive his paper.
There are 3 variants when 2 students doesn't receive their tests: {1, 3, 2}, {3, 2, 1}, {2, 1, 3}
There are 2 variants when 3 students doesn't receive their tests: {3, 1, 2}, {2, 3, 1}

There are 5 variants total and 6 overall possible situations. So the answer is (5 / 6) modulo 10^9 + 7 = 833333340


Given an integer n and a permutation of numbers 1, 2 ... , n-1, n write a program to print the permutation that lexicographically precedes the given input permutation. If the given permutation is the lexicographically least permutation, then print the input permutation itself.

Input Format:

First line is the test cases and second line contains value of integer n: 1 ≤ n ≤ 1,000,000
third line is a space separated list of integers 1 2 ... n permuted in some random order

Output Format:

Output a single line containing a space separated list of integers which is the lexicographically preceding permutation of the input permutation.

SAMPLE INPUT
1
3
1 3 2

SAMPLE OUTPUT
1 2 3