A11275 | Binary Path

You are given a $2 \times n$ grid filled with zeros and ones. Let the number at the intersection of the $i$ -th row and the $j$ -th column be $a_{ij}$ .

There is a grasshopper at the top-left cell $(1, 1)$ that can only jump one cell right or downwards. It wants to reach the bottom-right cell $(2, n)$ . Consider the binary string of length $n+1$ consisting of numbers written in cells of the path without changing their order.

Your goal is to:

1. Find the lexicographically smallest $^\dagger$ string you can attain by choosing any available path;
2. Find the number of paths that yield this lexicographically smallest string.

$^\dagger$ If two strings $s$ and $t$ have the same length, then $s$ is lexicographically smaller than $t$ if and only if in the first position where $s$ and $t$ differ, the string $s$ has a smaller element than the corresponding element in $t$ .

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 10^4$ ). The description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $2 \le n \le 2 \cdot 10^5$ ).

The second line of each test case contains a binary string $a_{11} a_{12} \ldots a_{1n}$ ( $a_{1i}$ is either $0$ or $1$ ).

The third line of each test case contains a binary string $a_{21} a_{22} \ldots a_{2n}$ ( $a_{2i}$ is either $0$ or $1$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$ .

For each test case, output two lines:

1. The lexicographically smallest string you can attain by choosing any available path;
2. The number of paths that yield this string.