Breaking news from zombie neurology! It turns out that – contrary to previous beliefs – every zombie is born with a single brain, and only later it evolves into a complicated brain structure. In fact, whenever a zombie consumes a brain, a new brain appears in its nervous system and gets immediately connected to one of the already existing brains using a single brain connector. Researchers are now interested in monitoring the brain latency of a zombie. Your task is to write a program which, given a history of evolution of a zombie's nervous system, computes its brain latency at every stage.

Input

The first line of the input contains one number n – the number of brains in the final nervous system (2 ≤ n ≤ 200000). In the second line a history of zombie's nervous system evolution is given. For convenience, we number all the brains by 1, 2, ..., n in the same order as they appear in the nervous system (the zombie is born with a single brain, number 1, and subsequently brains 2, 3, ..., n are added). The second line contains n - 1 space-separated numbers p2, p3, ..., pn, meaning that after a new brain k is added to the system, it gets connected to a parent-brain <image>.

Output

Output n - 1 space-separated numbers – the brain latencies after the brain number k is added, for k = 2, 3, ..., n.

Example

Input

6
1
2
2
1
5


Output

1 2 2 3 4 
#include <bits/stdc++.h>
using namespace std;
const int MX = 2e5 + 5;
long long pa[MX][64];
long long depth[MX], n;
int lca(int x, int y) {
  if (depth[y] > depth[x]) swap(x, y);
  if (depth[x] > depth[y]) {
    int h = depth[x] - depth[y];
    for (int i = 0; i < 31; i++) {
      if ((1 << i) & h) x = pa[x][i];
    }
  }
  for (int i = 31; i >= 0; i--) {
    if (pa[x][i] != pa[y][i]) {
      return lca(pa[x][i], pa[y][i]);
    }
  }
  if (x == y) return x;
  return pa[x][0];
  ;
}
int main() {
  cin >> n;
  int y, z, d;
  cin >> y;
  z = 2, d = 1;
  pa[1][0] = 1;
  depth[y] = 1;
  depth[z] = 2;
  pa[2][0] = 1;
  for (int i = 3; i <= n; i++) {
    int x;
    cin >> x;
    pa[i][0] = x;
    depth[i] = depth[x] + 1;
  }
  for (int j = 1; j < 30; j++) {
    for (int i = 1; i <= n; i++) {
      pa[i][j] = pa[pa[i][j - 1]][j - 1];
    }
  }
  cout << 1 << ' ';
  for (int i = 3; i <= n; i++) {
    int mx = depth[i] + depth[z] - 2 * depth[lca(z, i)];
    int fx = depth[i] + depth[y] - 2 * depth[lca(y, i)];
    if (mx > d) d = mx, y = i;
    if (fx > d) d = fx, z = i;
    cout << d << ' ';
  }
  return 0;
}
