311119: CF1937B. 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.

3
2
00
00
4
1101
1100
8
00100111
11101101

000
2
11000
1
001001101
4

In the first test case, the lexicographically smallest string is $\mathtt{000}$. There are two paths that yield this string:

In the second test case, the lexicographically smallest string is $\mathtt{11000}$. There is only one path that yields this string: