201083: [AtCoder]ARC108 D - AB

Problem Statement

Given are an integer $N$ and four characters $c_{\mathrm{AA}}$, $c_{\mathrm{AB}}$, $c_{\mathrm{BA}}$, and $c_{\mathrm{BB}}$. Here, it is guaranteed that each of those four characters is A or B.

Snuke has a string $s$, which is initially AB.

Let $|s|$ denote the length of $s$. Snuke can do the four kinds of operations below zero or more times in any order:

1. Choose $i$ such that $1 \leq i < |s|$, $s_{i}$ = A, $s_{i+1}$ = A and insert $c_{\mathrm{AA}}$ between the $i$-th and $(i+1)$-th characters of $s$.
2. Choose $i$ such that $1 \leq i < |s|$, $s_{i}$ = A, $s_{i+1}$ = B and insert $c_{\mathrm{AB}}$ between the $i$-th and $(i+1)$-th characters of $s$.
3. Choose $i$ such that $1 \leq i < |s|$, $s_{i}$ = B, $s_{i+1}$ = A and insert $c_{\mathrm{BA}}$ between the $i$-th and $(i+1)$-th characters of $s$.
4. Choose $i$ such that $1 \leq i < |s|$, $s_{i}$ = B, $s_{i+1}$ = B and insert $c_{\mathrm{BB}}$ between the $i$-th and $(i+1)$-th characters of $s$.

Find the number, modulo $(10^9+7)$, of strings that can be $s$ when Snuke has done the operations so that the length of $s$ becomes $N$.

Constraints

• $2 \leq N \leq 1000$
• Each of $c_{\mathrm{AA}}$, $c_{\mathrm{AB}}$, $c_{\mathrm{BA}}$, and $c_{\mathrm{BB}}$ is A or B.

Input

Input is given from Standard Input in the following format:

$N$
$c_{\mathrm{AA}}$
$c_{\mathrm{AB}}$
$c_{\mathrm{BA}}$
$c_{\mathrm{BB}}$

Output

Print the number, modulo $(10^9+7)$, of strings that can be $s$ when Snuke has done the operations so that the length of $s$ becomes $N$.

Sample Input 1

4
A
B
B
A

Sample Output 1

2

• There are two strings that can be $s$ when Snuke is done: ABAB and ABBB.

Sample Input 2

1000
B
B
B
B

Sample Output 2

1

• There is just one string that can be $s$ when Snuke is done.