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>
const int N = 200005;
using namespace std;
int n, a, b, c;
int P[N][18], L[N];
int LCA(int x, int y) {
  if (L[x] < L[y]) swap(x, y);
  for (int j = 17; j >= 0; j--)
    if (L[x] - (1 << j) >= L[y]) x = P[x][j];
  if (x == y) return x;
  for (int j = 17; j >= 0; j--)
    if (P[x][j] != P[y][j]) x = P[x][j], y = P[y][j];
  return P[x][0];
}
int dis(int x, int y) {
  int lca = LCA(x, y);
  return L[x] + L[y] - L[lca] - L[lca];
}
int main() {
  scanf("%d", &n);
  a = b = 1;
  for (int v = 2, p; v <= n; v++) {
    scanf("%d", &p);
    L[v] = L[p] + 1;
    P[v][0] = p;
    for (int j = 1; j < 18; j++) P[v][j] = P[P[v][j - 1]][j - 1];
    int x = dis(a, v);
    int y = dis(b, v);
    if (x >= y && c < x)
      b = v, c = x;
    else if (x < y && c < y)
      a = v, c = y;
    printf("%d ", c);
  }
}
