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## [Solution] Palindromic Deletions Solution Round C 2022 - Kick Start 2022

### Problem

Games with words and strings are very popular lately. Now Edsger tries to create a similar new game of his own. Here is what he came up with so far.

Edsger's new game is called Palindromic Deletions. As a player of this game, you are given a string of length $N$. Then you will perform the following process $N$ times:

1. Pick an index in the current string uniformly at random.
2. Delete the character at that index. You will then end up with a new string with one fewer character.
3. If the new string is a palindrome, you eat a piece of candy in celebration.

Now Edsger wonders: given a starting string, what is the expected number of candies you will eat during the game?

### Input

The first line of the input gives the number of test cases, $T$$T$ test cases follow. Each test case consists of two lines.

The first line of each test case contains an integer $N$, representing the length of the string.

The second line of each test case contains a string $S$ of length $N$, consisting of lowercase English characters.

### Output

For each test case, output one line containing Case #x$x$: E$E$, where $x$ is the test case number (starting from 1) and $E$ is the expected number of candies you will eat during the game.

$E$ should be computed modulo the prime ${10}^{9}+7$ ($1000000007$) as follows. Represent the answer of a test case as an irreducible fraction $\frac{p}{q}$. The number $E$ then must satisfy the modular equation $E×q\equiv p\phantom{\rule{0.444em}{0ex}}\left(\mathrm{mod}\phantom{\rule{0.333em}{0ex}}\left({10}^{9}+7\right)\right)$, and be between $0$ and ${10}^{9}+6$, inclusive. It can be shown that under the constraints of this problem, such a number $E$ always exists and can be uniquely determined.

### Limits

Time limit: 30 seconds.
Memory limit: 1 GB.
$1\le \mathbf{T}\le 20$.
String $S$ consists of only lowercase letters of the English alphabet.

#### Test Set 1

$2\le \mathbf{N}\le 8$.

#### Test Set 2

$2\le \mathbf{N}\le 400$.