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;
int d[200005], f[200005][20], n, i, j, x, y, z, z1, z2;
int LCA(int x, int y) {
  int i, j;
  if (d[x] > d[y]) x ^= y ^= x ^= y;
  if (d[x] != d[y])
    for (i = 19; i >= 0; i--)
      if (d[y] - d[x] >> i & 1) y = f[y][i];
  if (x != y) {
    for (i = 19; i >= 0; i--)
      if (f[x][i] != f[y][i]) {
        x = f[x][i];
        y = f[y][i];
      }
    x = f[x][0];
  }
  return x;
}
int dis(int x, int y) { return d[x] + d[y] - 2 * d[LCA(x, y)]; }
int main() {
  scanf("%d", &n);
  for (x = y = 1, i = 2; i <= n; i++) {
    scanf("%d", f[i]);
    d[i] = d[f[i][0]] + 1;
    for (j = 1; j < 20; j++) f[i][j] = f[f[i][j - 1]][j - 1];
    z1 = dis(i, x);
    z2 = dis(i, y);
    if (max(z1, z2) > z)
      if (z1 > z2) {
        y = i;
        z = z1;
      } else {
        x = i;
        z = z2;
      }
    printf("%d ", z);
  }
  return 0;
}
