Group Photo

1000ms

262144K

Description:

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:

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

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.

Input:

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 $$$200\,000$$$.

Output:

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

Sample Input:

Sample Output:

10
7
1

Note:

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.

Informação

Provedor Codeforces

Código CF1517E