A15417. Balance Addicts

Given an integer sequence $a_1, a_2, \dots, a_n$ of length $n$ , your task is to compute the number, modulo $998244353$ , of ways to partition it into several non-empty continuous subsequences such that the sums of elements in the subsequences form a balanced sequence.

A sequence $s_1, s_2, \dots, s_k$ of length $k$ is said to be balanced, if $s_{i} = s_{k-i+1}$ for every $1 \leq i \leq k$ . For example, $[1, 2, 3, 2, 1]$ and $[1,3,3,1]$ are balanced, but $[1,5,15]$ is not.

Formally, every partition can be described by a sequence of indexes $i_1, i_2, \dots, i_k$ of length $k$ with $1 = i_1 < i_2 < \dots < i_k \leq n$ such that

1. $k$ is the number of non-empty continuous subsequences in the partition;
2. For every $1 \leq j \leq k$ , the $j$ -th continuous subsequence starts with $a_{i_j}$ , and ends exactly before $a_{i_{j+1}}$ , where $i_{k+1} = n + 1$ . That is, the $j$ -th subsequence is $a_{i_j}, a_{i_j+1}, \dots, a_{i_{j+1}-1}$ .

There are $2^{n-1}$ different partitions in total. Let $s_1, s_2, \dots, s_k$ denote the sums of elements in the subsequences with respect to the partition $i_1, i_2, \dots, i_k$ . Formally, for every $1 \leq j \leq k$ , $$$$ s_j = \sum_{i=i_{j}}^{i_{j+1}-1} a_i = a_{i_j} + a_{i_j+1} + \dots + a_{i_{j+1}-1}. $$ For example, the partition $\[1\\,|\\,2,3\\,|\\,4,5,6\]$ of sequence $\[1,2,3,4,5,6\]$ is described by the sequence $\[1,2,4\]$ of indexes, and the sums of elements in the subsequences with respect to the partition is $\[1,5,15\]$ .</p><p>Two partitions $i\_1, i\_2, \\dots, i\_k$ and $i'\_1, i'\_2, \\dots, i'\_{k'}$ (described by sequences of indexes) are considered to be different, if at least one of the following holds. </p><ul> <li> $k \\neq k'$ , </li><li> $i\_j \\neq i'\_j$ for some $1 \\leq j \\leq \\min\\left\\{ k, k' \\right\\}$$$.

Each test contains multiple test cases. The first line contains an integer $t$ ( $1 \leq t \leq 10^5$ ) — the number of test cases. The following lines contain the description of each test case.

The first line of each test case contains an integer $n$ ( $1 \leq n \leq 10^5$ ), indicating the length of the sequence $a$ .

The second line of each test case contains $n$ integers $a_1, a_2, \dots, a_n$ ( $0 \leq a_i \leq 10^9$ ), indicating the elements of the sequence $a$ .

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

For each test case, output the number of partitions with respect to which the sum of elements in each subsequence is balanced, modulo $998244353$ .

For the first test case, there is only one way to partition a sequence of length $1$ , which is itself and is, of course, balanced.

For the second test case, there are $2$ ways to partition it:

- The sequence $[1, 1]$ itself, then $s = [2]$ is balanced;
- Partition into two subsequences $[1\,|\,1]$ , then $s = [1, 1]$ is balanced.

For the third test case, there are $3$ ways to partition it:

- The sequence $[0, 0, 1, 0]$ itself, then $s = [1]$ is balanced;
- $[0 \,|\, 0, 1 \,|\, 0]$ , then $s = [0, 1, 0]$ is balanced;
- $[0, 0 \,|\, 1 \,|\, 0]$ , then $s = [0, 1, 0]$ is balanced.

For the fourth test case, there are $4$ ways to partition it:

- The sequence $[1, 2, 3, 2, 1]$ itself, then $s = [9]$ is balanced;
- $[1, 2 \,|\, 3 \,|\, 2, 1]$ , then $s = [3, 3, 3]$ is balanced;
- $[1 \,|\, 2, 3, 2 \,|\, 1]$ , then $s = [1, 7, 1]$ is balanced;
- $[1 \,|\, 2 \,|\, 3 \,|\, 2 \,|\, 1]$ , then $s = [1, 2, 3, 2, 1]$ is balanced.

For the fifth test case, there are $2$ ways to partition it:

- The sequence $[1, 3, 5, 7, 9]$ itself, then $s = [25]$ is balanced;
- $[1, 3, 5 \,|\, 7 \,|\, 9]$ , then $s = [9, 7, 9]$ is balanced.

For the sixth test case, every possible partition should be counted. So the answer is $2^{32-1} \equiv 150994942 \pmod {998244353}$ .