F. Binary String Reconstruction
constructive algorithms; dfs and similar; math
For some binary string \(s\) (i.e. each character \(s_i\) is either '0' or '1'), all pairs of consecutive (adjacent) characters were written. In other words, all substrings of length \(2\) were written. For each pair (substring of length \(2\)), the number of '1' (ones) in it was calculated.You are given three numbers: \(n_0\) — the number of such pairs of consecutive characters (substrings) where the number of ones equals \(0\); \(n_1\) — the number of such pairs of consecutive characters (substrings) where the number of ones equals \(1\); \(n_2\) — the number of such pairs of consecutive characters (substrings) where the number of ones equals \(2\). For example, for the string \(s=\)""1110011110"", the following substrings would be written: ""11"", ""11"", ""10"", ""00"", ""01"", ""11"", ""11"", ""11"", ""10"". Thus, \(n_0=1\), \(n_1=3\), \(n_2=5\).Your task is to restore any suitable binary string \(s\) from the given values \(n_0, n_1, n_2\). It is guaranteed that at least one of the numbers \(n_0, n_1, n_2\) is greater than \(0\). Also, it is guaranteed that a solution exists.
The first line contains an integer \(t\) (\(1 \le t \le 1000\)) — the number of test cases in the input. Then test cases follow.Each test case consists of one line which contains three integers \(n_0, n_1, n_2\) (\(0 \le n_0, n_1, n_2 \le 100\); \(n_0 + n_1 + n_2 > 0\)). It is guaranteed that the answer for given \(n_0, n_1, n_2\) exists.
Print \(t\) lines. Each of the lines should contain a binary string corresponding to a test case. If there are several possible solutions, print any of them.