#P1762C. Binary Strings are Fun
Binary Strings are Fun
Description
A binary string$^\dagger$ $b$ of odd length $m$ is good if $b_i$ is the median$^\ddagger$ of $b[1,i]^\S$ for all odd indices $i$ ($1 \leq i \leq m$).
For a binary string $a$ of length $k$, a binary string $b$ of length $2k-1$ is an extension of $a$ if $b_{2i-1}=a_i$ for all $i$ such that $1 \leq i \leq k$. For example, 1001011 and 1101001 are extensions of the string 1001. String $x=$1011011 is not an extension of string $y=$1001 because $x_3 \neq y_2$. Note that there are $2^{k-1}$ different extensions of $a$.
You are given a binary string $s$ of length $n$. Find the sum of the number of good extensions over all prefixes of $s$. In other words, find $\sum_{i=1}^{n} f(s[1,i])$, where $f(x)$ gives number of good extensions of string $x$. Since the answer can be quite large, you only need to find it modulo $998\,244\,353$.
$^\dagger$ A binary string is a string whose elements are either $\mathtt{0}$ or $\mathtt{1}$.
$^\ddagger$ For a binary string $a$ of length $2m-1$, the median of $a$ is the (unique) element that occurs at least $m$ times in $a$.
$^\S$ $a[l,r]$ denotes the string of length $r-l+1$ which is formed by the concatenation of $a_l,a_{l+1},\ldots,a_r$ in that order.
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 $n$ ($1 \le n \le 2 \cdot 10^5$), where $n$ is the length of the binary string $s$.
The second line of each test case contains the binary string $s$ of length $n$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
For each test case, print the answer modulo $998\,244\,353$.
Input
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 $n$ ($1 \le n \le 2 \cdot 10^5$), where $n$ is the length of the binary string $s$.
The second line of each test case contains the binary string $s$ of length $n$.
It is guaranteed that the sum of $n$ over all test cases does not exceed $2 \cdot 10^5$.
Output
For each test case, print the answer modulo $998\,244\,353$.
6
1
1
1
0
2
11
3
010
9
101101111
37
1011011111011010000011011111111011111
1
1
3
3
21
365
Note
In the first and second test cases, $f(s[1,1])=1$.
In the third test case, the answer is $f(s[1,1])+f(s[1,2])=1+2=3$.
In the fourth test case, the answer is $f(s[1,1])+f(s[1,2])+f(s[1,3])=1+1+1=3$.
$f(\mathtt{11})=2$ because two good extensions are possible: 101 and 111.
$f(\mathtt{01})=1$ because only one good extension is possible: 011.