CF1613D MEX Sequences

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).

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 $ .

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

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.