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 mxN = 2e5;
int n, d[mxN], anc[mxN][18];
array<int, 3> b;
int lca(int u, int v) {
  if (d[u] > d[v]) swap(u, v);
  for (int i = 17; ~i; --i)
    if (d[u] <= d[v] - (1 << i)) v = anc[v][i];
  if (u == v) return u;
  for (int i = 17; ~i; --i)
    if (anc[u][i] ^ anc[v][i]) u = anc[u][i], v = anc[v][i];
  return anc[u][0];
}
int di(int u, int v) { return d[u] + d[v] - 2 * d[lca(u, v)]; }
int main() {
  ios::sync_with_stdio(0);
  cin.tie(0);
  cin >> n;
  for (int i = 1; i < n; ++i) {
    cin >> anc[i][0], --anc[i][0];
    d[i] = d[anc[i][0]] + 1;
    for (int j = 1; j < 18; ++j) anc[i][j] = anc[anc[i][j - 1]][j - 1];
    int x = b[1], y = b[2];
    b = max(b, {di(i, x), i, x});
    b = max(b, {di(i, y), i, y});
    cout << b[0] << " ";
  }
  return 0;
}
