Problem: Yuezheng Ling gives Luo Tianyi a tree which has n nodes, rooted at 1. 

Luo Tianyi will tell you that the parent of the i-th node is a_i (1 ≤ a_i<i for 2 ≤ i ≤ n), and she will ask you to perform q queries of 2 types:

  1. She'll give you three integers l, r and x (2 ≤ l ≤ r ≤ n, 1 ≤ x ≤ 10^5). You need to replace a_i with max(a_i-x,1) for all i with l ≤ i ≤ r.
  2. She'll give you two integers u, v (1 ≤ u, v ≤ n). You need to find the [LCA](https://en.wikipedia.org/wiki/Lowest_common_ancestor) of nodes u and v (their lowest common ancestor).

Input

The first line contains two integers n and q (2≤ n,q ≤ 10^5) — the number of nodes and the number of queries, respectively.

The second line contains n-1 integers a_2, a_3,..., a_n (1 ≤ a_i < i), where a_i is the parent of the node i.

Next q lines contain queries. For each query, the first integer of each line is t (t = 1 or 2) — the type of the query. 

If t = 1, this represents the query of the first type. Then, three integers will follow: l, r, x (2 ≤ l ≤ r ≤ n, 1 ≤ x ≤ 10^5), meaning that you have to replace a_i with max(a_i-x,1) for all i with l ≤ i ≤ r.

If t = 2, this represents the query of the second type. Then, two integers will follow: u and v (1 ≤ u, v ≤ n), and you have to find the LCA of u and v. 

It's guaranteed that there is at least one query of the second type.

Output

For each query of the second type output answer on a new line.

Example

Input


6 4
1 2 3 3 4
2 3 4
1 2 3 1
2 5 6
2 2 3


Output


3
3
1

Note

The tree in example is shown below.

<image>

After the query of the first type, the tree changes and is looking as shown below.

<image>

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
#ifdef ALGO
#include "el_psy_congroo.hpp"
#else
#define DUMP(...) 1145141919810
#define CHECK(...) (__VA_ARGS__)
#endif

const int kN = 100000 + 5;
const int BIT = 7;
const int MAGIC = 1 << BIT;
int n, q;
int a[kN];
int jump[MAGIC][kN];
int color[MAGIC][kN];
int offset[kN / MAGIC + 1];

bool touched[kN / MAGIC + 1][MAGIC];

inline int get_block_id(int p) { return p >> BIT; }

void prepare(int b) {
  int l = b * MAGIC;
  int r = std::min(n - 1, (b + 1) * MAGIC - 1);
  for (int x = offset[b]; x < MAGIC; ++x) if (x == 0 || touched[b][x]) {
    int c = 1;
    for (int i = l; i <= r; ++i) {
      int w = std::max(0, a[i] - x);
      if (w < l) {
        color[x][i] = c++;
        jump[x][i] = w;
      } else {
        color[x][i] = color[x][w];
        jump[x][i] = jump[x][w];
      }
    }
  }
}

void update(int l, int r, int x) {
  int lb = get_block_id(l);
  int rb = get_block_id(r);
  for (int b = lb + 1; b < rb; ++b) {
    offset[b] = std::min(offset[b] + x, kN);
  }
  for (int i = l; i < (lb + 1) * MAGIC && i <= r; ++i) {
    a[i] = std::max(0, a[i] - x);
  }
  prepare(lb);
  if (lb != rb) {
    for (int i = rb * MAGIC; i <= r; ++i) {
      a[i] = std::max(0, a[i] - x);
    }
    prepare(rb);
  }
}

int jump_one_step(int u) {
  return std::max(0, a[u] - offset[get_block_id(u)]);
}

int jump_one_block(int u) {
  if (offset[get_block_id(u)] >= MAGIC) return jump_one_step(u);
  return jump[offset[get_block_id(u)]][u];
}

int get_lca(int u, int v) {
  while (true) {
    if (u > v) std::swap(u, v);
    // u <= v
    if (u == v) {
      return u;
    } else if (get_block_id(u) != get_block_id(v)) {
      v = jump_one_block(v);
    } else if (offset[get_block_id(v)] >= MAGIC) {
      v = jump_one_step(v);
    } else if (color[offset[get_block_id(u)]][u] == color[offset[get_block_id(v)]][v]) {
      v = jump_one_step(v);
    } else {
      v = jump_one_block(v);
    }
  }
}

int t[kN], l[kN], r[kN], x[kN], u[kN], v[kN];

int main() {
  std::ios::sync_with_stdio(false);
  std::cin.tie(nullptr);
  std::istream& reader = std::cin;

  reader >> n >> q;
  for (int i = 1; i < n; ++i) {
    reader >> a[i]; --a[i];
  }
  for (int i = 0; i < q; ++i) {
    reader >> t[i];
    if (t[i] == 1) {
      reader >> l[i] >> r[i] >> x[i]; --l[i]; --r[i];
      for (int b = get_block_id(l[i]) + 1; b < get_block_id(r[i]); ++b) {
        offset[b] = std::min(offset[b] + x[i], kN);
        if (offset[b] < MAGIC)
          touched[b][offset[b]] = true;
      }
    } else {
      reader >> u[i] >> v[i]; --u[i]; --v[i];
    }
  }
  memset(offset, 0, sizeof(offset));
  for (int b = 0; b * MAGIC < n; ++b) {
    prepare(b);
  }
  for (int i = 0; i < q; ++i) {
    if (t[i] == 1) {
      update(l[i], r[i], x[i]);
    } else {
      std::cout << (get_lca(u[i], v[i]) + 1) << std::endl;
    }
  }
}