1613D · MEX Sequences

Problem: Let's call a sequence of integers $x_1, x_2, \dots, x_k$ MEX-correct if for all $i$ ($1 \le i \le k$) $|x_i - \operatorname{MEX}(x_1, x_2, \dots, x_i)| \le 1$ holds. Where $\operatorname{MEX}(x_1, \dots, x_k)$ is the minimum non-negative integer that doesn't belong to the set $x_1, \dots, x_k$. For example, $\operatorname{MEX}(1, 0, 1, 3) = 2$ and $\operatorname{MEX}(2, 1, 5) = 0$.

You are given an array $a$ consisting of $n$ non-negative integers. Calculate the number of non-empty MEX-correct subsequences of a given array. The number of subsequences can be very large, so print it modulo $998244353$.

Note: a subsequence of an array $a$ is a sequence $[a_{i_1}, a_{i_2}, \dots, a_{i_m}]$ meeting the constraints $1 \le i_1 < i_2 < \dots < i_m \le n$. If two different ways to choose the sequence of indices $[i_1, i_2, \dots, i_m]$ yield the same subsequence, the resulting subsequence should be counted twice (i. e. two subsequences are different if their sequences of indices $[i_1, i_2, \dots, i_m]$ are not the same).

Input Format: The first line contains a single integer $t$ ($1 \le t \le 10^5$) — the number of test cases.

The first line of each test case contains a single integer $n$ ($1 \le n \le 5 \cdot 10^5$).

The second line contains $n$ integers $a_1, a_2, \dots, a_n$ ($0 \le a_i \le n$).

The sum of $n$ over all test cases doesn't exceed $5 \cdot 10^5$.

Output Format: For each test case, print a single integer — the number of non-empty MEX-correct subsequences of a given array, taken modulo $998244353$.

Note: In the first example, the valid subsequences are $[0]$, $[1]$, $[0,1]$ and $[0,2]$.

In the second example, the valid subsequences are $[0]$ and $[1]$.

In the third example, any non-empty subsequence is valid.