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;
vector<vector<long long>> p;
long long n;
long long a;
long long b;
vector<long long> d;
long long ans = 1;
long long lca(long long a, long long b) {
  if (d[b] > d[a]) {
    swap(a, b);
  }
  long long x = d[a] - d[b];
  for (long long i = 0; i < 18; i++) {
    if ((x >> i) & 1) {
      a = p[a][i];
    }
  }
  if (a == b) {
    return a;
  }
  for (long long i = 17; i >= 0; i--) {
    if (p[a][i] != p[b][i]) {
      a = p[a][i];
      b = p[b][i];
    }
  }
  return p[a][0];
}
signed main() {
  cin >> n;
  p.resize(n + 1, vector<long long>(18));
  d.resize(n + 1);
  d[1] = 0;
  d[2] = 1;
  long long p1;
  cin >> p1;
  a = 1;
  b = 2;
  p[2][0] = 1;
  cout << 1 << " ";
  for (long long i = 3; i <= n; i++) {
    cin >> p1;
    p[i][0] = p1;
    for (long long j = 1; j < 18; j++) {
      p[i][j] = p[p[i][j - 1]][j - 1];
    }
    d[i] = d[p1] + 1;
    long long v = lca(i, a);
    if (d[i] - d[v] + d[a] - d[v] > ans) {
      ans = d[i] - d[v] + d[a] - d[v];
      b = i;
    } else {
      v = lca(i, b);
      if (d[i] - d[v] + d[b] - d[v] > ans) {
        ans = d[i] - d[v] + d[b] - d[v];
        a = i;
      }
    }
    cout << ans << " ";
  }
}
