A13730 | Binary String Reconstruction

Consider the following process. You have a binary string (a string where each character is either 0 or 1) $w$ of length $n$ and an integer $x$ . You build a new binary string $s$ consisting of $n$ characters. The $i$ -th character of $s$ is chosen as follows:

- if the character $w_{i-x}$ exists and is equal to 1, then $s_i$ is 1 (formally, if $i > x$ and $w_{i-x} = $ 1, then $s_i = $ 1);
- if the character $w_{i+x}$ exists and is equal to 1, then $s_i$ is 1 (formally, if $i + x \le n$ and $w_{i+x} = $ 1, then $s_i = $ 1);
- if both of the aforementioned conditions are false, then $s_i$ is 0.

You are given the integer $x$ and the resulting string $s$ . Reconstruct the original string $w$ .

The first line contains one integer $t$ ( $1 \le t \le 1000$ ) — the number of test cases.

Each test case consists of two lines. The first line contains the resulting string $s$ ( $2 \le |s| \le 10^5$ , each character of $s$ is either 0 or 1). The second line contains one integer $x$ ( $1 \le x \le |s| - 1$ ).

The total length of all strings $s$ in the input does not exceed $10^5$ .

For each test case, print the answer on a separate line as follows:

- if no string $w$ can produce the string $s$ at the end of the process, print $-1$ ;
- otherwise, print the binary string $w$ consisting of $|s|$ characters. If there are multiple answers, print any of them.