E2. Close Tuples (hard version)
binary search; combinatorics; implementation; math; sortings; two pointers
This is the hard version of this problem. The only difference between the easy and hard versions is the constraints on \(k\) and \(m\). In this version of the problem, you need to output the answer by modulo \(10^9+7\).You are given a sequence \(a\) of length \(n\) consisting of integers from \(1\) to \(n\). The sequence may contain duplicates (i.e. some elements can be equal).Find the number of tuples of \(m\) elements such that the maximum number in the tuple differs from the minimum by no more than \(k\). Formally, you need to find the number of tuples of \(m\) indices \(i_1 < i_2 < \ldots < i_m\), such that$$$\(\max(a_{i_1}, a_{i_2}, \ldots, a_{i_m}) - \min(a_{i_1}, a_{i_2}, \ldots, a_{i_m}) \le k.\)\(For example, if \)n=4\(, \)m=3\(, \)k=2\(, \)a=[1,2,4,3]\(, then there are two such triples (\)i=1, j=2, z=4\( and \)i=2, j=3, z=4\(). If \)n=4\(, \)m=2\(, \)k=1\(, \)a=[1,1,1,1]\(, then all six possible pairs are suitable.As the result can be very large, you should print the value modulo \)10^9 + 7\( (the remainder when divided by \)10^9 + 7$$$).
The first line contains a single integer \(t\) (\(1 \le t \le 2 \cdot 10^5\)) — the number of test cases. Then \(t\) test cases follow.The first line of each test case contains three integers \(n\), \(m\), \(k\) (\(1 \le n \le 2 \cdot 10^5\), \(1 \le m \le 100\), \(1 \le k \le n\)) — the length of the sequence \(a\), number of elements in the tuples and the maximum difference of elements in the tuple.The next line contains \(n\) integers \(a_1, a_2,\ldots, a_n\) (\(1 \le a_i \le n\)) — the sequence \(a\).It is guaranteed that the sum of \(n\) for all test cases does not exceed \(2 \cdot 10^5\).
Output \(t\) answers to the given test cases. Each answer is the required number of tuples of \(m\) elements modulo \(10^9 + 7\), such that the maximum value in the tuple differs from the minimum by no more than \(k\).