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 N = 200005;
int fa[N][19];
int dep[N];
int tx, ty, ta1, ta2;
int ans, n;
int lca(int x, int y) {
  if (dep[x] < dep[y]) swap(x, y);
  for (int j = 18; j >= 0; j--)
    if (dep[fa[x][j]] >= dep[y]) x = fa[x][j];
  if (x == y) return x;
  int ans;
  for (int j = 18; j >= 0; j--)
    if (fa[x][j] != fa[y][j]) x = fa[x][j], y = fa[y][j];
  return fa[x][0];
}
int main() {
  scanf("%d", &n);
  fa[1][0] = 0;
  dep[1] = 1;
  tx = 1;
  ty = 1;
  for (int i = 2; i <= n; i++) {
    int x, y;
    scanf("%d", &x);
    fa[i][0] = x;
    dep[i] = dep[x] + 1;
    for (int j = 1; j <= 18; j++) fa[i][j] = fa[fa[i][j - 1]][j - 1];
    ta1 = lca(tx, i);
    ta2 = lca(ty, i);
    x = tx;
    y = ty;
    if (-(dep[ta1] * 2 - dep[tx] - dep[i]) > ans) {
      ans = -(dep[ta1] * 2 - dep[tx] - dep[i]);
      x = tx, y = i;
    }
    if (-(dep[ta2] * 2 - dep[ty] - dep[i]) > ans) {
      ans = -(dep[ta2] * 2 - dep[ty] - dep[i]);
      y = ty, x = i;
    }
    tx = x, ty = y;
    printf("%d ", ans);
  }
  return 0;
}
