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 inf = 1e9;
const int MOD = 1e9 + 7;
const int N = 2e5 + 10;
int n, pa[N][20], lv[N];
void prepare() {
  scanf("%d", &n);
  for (auto i = (2); i <= (n); i++) scanf("%d", pa[i]);
  for (auto i = (2); i <= (n); i++) lv[i] = lv[pa[i][0]] + 1;
  for (auto lay = (1); lay <= (19); lay++)
    for (auto i = (1); i <= (n); i++) pa[i][lay] = pa[pa[i][lay - 1]][lay - 1];
}
int LCA(int x, int y) {
  if (lv[x] < lv[y]) swap(x, y);
  for (int i = 0, val = lv[x] - lv[y]; val; i++, val >>= 1)
    if (val & 1) x = pa[x][i];
  if (x == y) return x;
  for (int i = int(log2(lv[x])); i >= 0; i--)
    if (pa[x][i] != pa[y][i]) x = pa[x][i], y = pa[y][i];
  return pa[x][0];
}
int dist(int x, int y) { return lv[x] + lv[y] - 2 * lv[LCA(x, y)]; }
void solve() {
  int s1 = 1, s2 = 1;
  for (auto i = (2); i <= (n); i++) {
    int ds1 = dist(i, s1), ds2 = dist(i, s2);
    if (max(ds1, ds2) > dist(s1, s2)) {
      if (ds1 > ds2)
        s2 = i;
      else
        s1 = i;
    }
    printf("%d ", dist(s1, s2));
  }
}
int main() {
  prepare();
  solve();
}
