CF1517E.Group Photo

In the 2050 Conference, some people from the competitive programming community meet together and are going to take a photo. The $n$ people form a line. They are numbered from $1$ to $n$ from left to right. Each of them either holds a cardboard with the letter 'C' or a cardboard with the letter 'P'.

Let $C=\{c_1,c_2,\dots,c_m\}$ $(c_1<c_2<\ldots <c_m)$ be the set of people who hold cardboards of 'C'. Let $P=\{p_1,p_2,\dots,p_k\}$ $(p_1<p_2<\ldots <p_k)$ be the set of people who hold cardboards of 'P'. The photo is good if and only if it satisfies the following constraints:

1. $C\cup P=\{1,2,\dots,n\}$
2. $C\cap P =\emptyset $ .
3. $c_i-c_{i-1}\leq c_{i+1}-c_i(1< i <m)$ .
4. $p_i-p_{i-1}\geq p_{i+1}-p_i(1< i <k)$ .

Given an array $a_1,\ldots, a_n$ , please find the number of good photos satisfying the following condition: $$$$\sum\limits_{x\in C} a_x < \sum\limits_{y\in P} a_y. $$

The answer can be large, so output it modulo $998\\,244\\,353$$$. Two photos are different if and only if there exists at least one person who holds a cardboard of 'C' in one photo but holds a cardboard of 'P' in the other.

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 200\,000$ ). Description of the test cases follows.

The first line of each test case contains a single integer $n$ ( $1\leq n\leq 200\,000$ ).

The second line contains $n$ integers $a_1, a_2, \ldots, a_n$ ( $1 \le a_i \le 10^9$ ).

It is guaranteed that the sum of $n$ over all test cases does not exceed $200\,000$ .

For each test case, output the answer modulo $998\,244\,353$ in a separate line.

For the first test case, there are $10$ possible good photos satisfying the condition: PPPPP, CPPPP, PCPPP, CCPPP, PCCPP, PCPCP, PPPPC, CPPPC, PCPPC, PPPCC.

For the second test case, there are $7$ possible good photos satisfying the condition: PPPP, PCPP, PCCP, PPPC, PCPC, PPCC, PCCC.