description: Let's call a sequence of integers x_1, x_2, ..., x_k MEX-correct if for all i (1 ≤ i ≤ k) |x_i - \operatorname{MEX}(x_1, x_2, ..., x_i)| ≤ 1 holds. Where \operatorname{MEX}(x_1, ..., x_k) is the minimum non-negative integer that doesn't belong to the set x_1, ..., 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}, ..., a_{i_m}] meeting the constraints 1 ≤ i_1 < i_2 < ... < i_m ≤ n. If two different ways to choose the sequence of indices [i_1, i_2, ..., 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, ..., i_m] are not the same).
  
  Input
  
  The first line contains a single integer t (1 ≤ t ≤ 10^5) — the number of test cases.
  
  The first line of each test case contains a single integer n (1 ≤ n ≤ 5 ⋅ 10^5).
  
  The second line contains n integers a_1, a_2, ..., a_n (0 ≤ a_i ≤ n).
  
  The sum of n over all test cases doesn't exceed 5 ⋅ 10^5.
  
  Output
  
  For each test case, print a single integer — the number of non-empty MEX-correct subsequences of a given array, taken modulo 998244353.
  
  Example
  
  Input
  
  
  4
  3
  0 2 1
  2
  1 0
  5
  0 0 0 0 0
  4
  0 1 2 3
  
  
  Output
  
  
  4
  2
  31
  7
  
  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. 
solution: #include <bits/stdc++.h>
  using namespace std;
  mt19937 rng(chrono::steady_clock::now().time_since_epoch().count());
  const long long N = 1e6 + 10, inf = 1e9, mod = 998244353;
  long long a[N + 2], dp[N + 2], dpSum[N + 2], dpRev[N + 2];
  vector<long long> po[N + 2];
  long long howMany(long long x, long long id) {
    if (x < 0) return 0;
    long long i = lower_bound(po[x].begin(), po[x].end(), id) - po[x].begin();
    return (long long)po[x].size() - i;
  }
  long long pw[N + 2];
  int main() {
    ios_base::sync_with_stdio(0);
    cin.tie(0);
    pw[0] = 1;
    for (long long i = 1; i <= N; i++) pw[i] = (pw[i - 1] * 2) % mod;
    long long t;
    cin >> t;
    while (t--) {
      long long n;
      cin >> n;
      for (long long i = 0; i <= n + 4; i++) po[i].clear();
      for (long long i = 1; i <= n; i++) {
        cin >> a[i];
        po[a[i]].push_back(i);
      }
      for (long long i = 0; i <= n + 4; i++) dp[i] = 0;
      for (long long i = 0; i <= n + 4; i++) dpSum[i] = 0;
      for (long long i = 0; i <= n + 4; i++) dpRev[i] = 0;
      for (long long i = 1; i <= n; i++) {
        if (a[i] == 0) {
          dp[i] = (dpSum[0] + 1) % mod;
          dpSum[0] = (dpSum[0] + dp[i]) % mod;
        } else {
          dp[i] = (dpSum[a[i]] + dpSum[a[i] - 1]) % mod;
          dpSum[a[i]] = (dpSum[a[i]] + dp[i]) % mod;
        }
      }
      long long ans = 0;
      for (long long i = n; i >= 1; i--) {
        long long num = howMany(a[i], i + 1) + howMany(a[i] - 2, i + 1);
        dpRev[a[i]] = (dpRev[a[i]] + pw[num]) % mod;
        ans = (ans + 1LL * dp[i] * (dpRev[a[i] + 2] + 1)) % mod;
      }
      ans = (ans + dpRev[1]) % mod;
      cout << ans << endl;
    }
    return 0;
  }
  