Description:
You are given a binary string of length $$$n$$$. You have exactly $$$k$$$ moves. In one move, you must select a single bit. The state of all bits except that bit will get flipped ($$$0$$$ becomes $$$1$$$, $$$1$$$ becomes $$$0$$$). You need to output the lexicographically largest string that you can get after using all $$$k$$$ moves. Also, output the number of times you will select each bit. If there are multiple ways to do this, you may output any of them.
A binary string $$$a$$$ is lexicographically larger than a binary string $$$b$$$ of the same length, if and only if the following holds:
- in the first position where $$$a$$$ and $$$b$$$ differ, the string $$$a$$$ contains a $$$1$$$, and the string $$$b$$$ contains a $$$0$$$.
Input Format:
The first line contains a single integer $$$t$$$ ($$$1 \le t \le 1000$$$) — the number of test cases.
Each test case has two lines. The first line has two integers $$$n$$$ and $$$k$$$ ($$$1 \leq n \leq 2 \cdot 10^5$$$; $$$0 \leq k \leq 10^9$$$).
The second line has a binary string of length $$$n$$$, each character is either $$$0$$$ or $$$1$$$.
The sum of $$$n$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.
Output Format:
For each test case, output two lines.
The first line should contain the lexicographically largest string you can obtain.
The second line should contain $$$n$$$ integers $$$f_1, f_2, \ldots, f_n$$$, where $$$f_i$$$ is the number of times the $$$i$$$-th bit is selected. The sum of all the integers must be equal to $$$k$$$.
Note:
Here is the explanation for the first testcase. Each step shows how the binary string changes in a move.
- Choose bit $$$1$$$: $$$\color{red}{\underline{1}00001} \rightarrow \color{red}{\underline{1}}\color{blue}{11110}$$$.
- Choose bit $$$4$$$: $$$\color{red}{111\underline{1}10} \rightarrow \color{blue}{000}\color{red}{\underline{1}}\color{blue}{01}$$$.
- Choose bit $$$4$$$: $$$\color{red}{000\underline{1}01} \rightarrow \color{blue}{111}\color{red}{\underline{1}}\color{blue}{10}$$$.