#include <bits/stdc++.h>
using namespace std;
template <typename T>
struct BottomUpSegmentTree {
  int n;
  int N;
  vector<T> tree;
  T (*merge)(T, T);
  T defaultValue;
  BottomUpSegmentTree(int n, T defaultValue, T (*merge)(T, T)) {
    this->n = n;
    this->N = 2 * n;
    this->merge = merge;
    this->defaultValue = defaultValue;
    tree = vector<T>(N, defaultValue);
  }
  BottomUpSegmentTree(vector<T>& arr, T defaultValue, T (*merge)(T, T))
      : BottomUpSegmentTree(arr.size(), defaultValue, merge) {
    build(arr);
  }
  void build(vector<T>& arr) {
    for (int i = 0; i < n; i++) tree[n + i] = arr[i];
    for (int i = n - 1; i > 0; --i)
      tree[i] = merge(tree[i << 1], tree[i << 1 | 1]);
  }
  void updateTreeNode(int p, T value) {
    tree[p + n] = value;
    p = p + n;
    for (int i = p; i > 1; i >>= 1) tree[i >> 1] = merge(tree[i], tree[i ^ 1]);
  }
  T query(int l, int r) {
    T res = defaultValue;
    r++;
    for (l += n, r += n; l < r; l >>= 1, r >>= 1) {
      if (l & 1) res = merge(res, tree[l++]);
      if (r & 1) res = merge(res, tree[--r]);
    }
    return res;
  }
};
int merge(int x1, int x2) { return max(x1, x2); }
int main() {
  int n;
  scanf("%d", &n);
  vector<int> nums(n);
  int cnt = 0;
  for (int i = 0; i < n; i++) {
    scanf("%d", &nums[i]);
    if (nums[i] - i - 1 <= 0) cnt++;
  }
  vector<pair<int, int>> arr(cnt);
  cnt = 0;
  for (int i = 0; i < n; i++) {
    if (nums[i] - i - 1 <= 0) {
      arr[cnt] = {nums[i], -i};
      cnt++;
    }
  }
  vector<int> dp(cnt, 1);
  int ans = 0;
  sort(arr.begin(), arr.end());
  BottomUpSegmentTree<int> tree(n, 0, merge);
  for (int i = 0; i < cnt; i++) {
    int index = -arr[i].second - arr[i].first + 1;
    int val;
    if (index >= 0)
      val = tree.query(0, index);
    else
      val = 0;
    if (dp[i] <= val) dp[i] = val + 1;
    ans = max(ans, dp[i]);
    tree.updateTreeNode(index, dp[i]);
  }
  printf("%d", ans);
  return 0;
}

### Prompt

Please formulate a CPP solution to the following problem:

Mr. Chanek gives you a sequence a indexed from 1 to n. Define f(a) as the number of indices where a_i = i. 

You can pick an element from the current sequence and remove it, then concatenate the remaining elements together. For example, if you remove the 3-rd element from the sequence [4, 2, 3, 1], the resulting sequence will be [4, 2, 1]. 

You want to remove some elements from a in order to maximize f(a), using zero or more operations. Find the largest possible f(a).

Input

The first line contains one integer n (1 ≤ n ≤ 2 ⋅ 10^5) — the initial length of the sequence.

The second line contains n integers a_1, a_2, …, a_n (1 ≤ a_i ≤ 2 ⋅ 10^5) — the initial sequence a.

Output

Output an integer denoting the largest f(a) that can be obtained by doing zero or more operations.

Examples

Input


7
2 1 4 2 5 3 7


Output


3


Input


4
4 2 3 1


Output


2

Note

In the first example, f(A) = 3 by doing the following operations.

[2,1,4,2,5,3,7] → [2,1,2,5,3,7] → [1,2,5,3,7] → [1,2,5,3] → [1,2,3]

In the second example, f(A) = 2 and no additional operation is needed.

### Response

```cpp
#include <bits/stdc++.h>
using namespace std;
template <typename T>
struct BottomUpSegmentTree {
  int n;
  int N;
  vector<T> tree;
  T (*merge)(T, T);
  T defaultValue;
  BottomUpSegmentTree(int n, T defaultValue, T (*merge)(T, T)) {
    this->n = n;
    this->N = 2 * n;
    this->merge = merge;
    this->defaultValue = defaultValue;
    tree = vector<T>(N, defaultValue);
  }
  BottomUpSegmentTree(vector<T>& arr, T defaultValue, T (*merge)(T, T))
      : BottomUpSegmentTree(arr.size(), defaultValue, merge) {
    build(arr);
  }
  void build(vector<T>& arr) {
    for (int i = 0; i < n; i++) tree[n + i] = arr[i];
    for (int i = n - 1; i > 0; --i)
      tree[i] = merge(tree[i << 1], tree[i << 1 | 1]);
  }
  void updateTreeNode(int p, T value) {
    tree[p + n] = value;
    p = p + n;
    for (int i = p; i > 1; i >>= 1) tree[i >> 1] = merge(tree[i], tree[i ^ 1]);
  }
  T query(int l, int r) {
    T res = defaultValue;
    r++;
    for (l += n, r += n; l < r; l >>= 1, r >>= 1) {
      if (l & 1) res = merge(res, tree[l++]);
      if (r & 1) res = merge(res, tree[--r]);
    }
    return res;
  }
};
int merge(int x1, int x2) { return max(x1, x2); }
int main() {
  int n;
  scanf("%d", &n);
  vector<int> nums(n);
  int cnt = 0;
  for (int i = 0; i < n; i++) {
    scanf("%d", &nums[i]);
    if (nums[i] - i - 1 <= 0) cnt++;
  }
  vector<pair<int, int>> arr(cnt);
  cnt = 0;
  for (int i = 0; i < n; i++) {
    if (nums[i] - i - 1 <= 0) {
      arr[cnt] = {nums[i], -i};
      cnt++;
    }
  }
  vector<int> dp(cnt, 1);
  int ans = 0;
  sort(arr.begin(), arr.end());
  BottomUpSegmentTree<int> tree(n, 0, merge);
  for (int i = 0; i < cnt; i++) {
    int index = -arr[i].second - arr[i].first + 1;
    int val;
    if (index >= 0)
      val = tree.query(0, index);
    else
      val = 0;
    if (dp[i] <= val) dp[i] = val + 1;
    ans = max(ans, dp[i]);
    tree.updateTreeNode(index, dp[i]);
  }
  printf("%d", ans);
  return 0;
}

```