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 = 200000 + 10;
int n;
std::vector<int> adj[N];
int dep[N], anc[21][N], *fa = anc[0];
int lca(int x, int y) {
  if (dep[x] < dep[y]) std::swap(x, y);
  for (int i = 20; i >= 0; --i)
    if (dep[anc[i][x]] >= dep[y]) x = anc[i][x];
  if (x == y) return x;
  for (int i = 20; i >= 0; --i)
    if (anc[i][x] != anc[i][y]) x = anc[i][x], y = anc[i][y];
  return fa[x];
}
inline int dist(int x, int y) { return dep[x] + dep[y] - 2 * dep[lca(x, y)]; }
struct Info {
  int x, y, z;
  Info() {}
  Info(int _x, int _y, int _z) : x(_x), y(_y), z(_z) {}
  inline bool operator<(const Info &rhs) const { return z < rhs.z; }
};
int main() {
  scanf("%d", &n);
  dep[1] = 1;
  for (int i = 2; i <= n; ++i) scanf("%d", &fa[i]), dep[i] = dep[fa[i]] + 1;
  for (int i = 1; i < 21; ++i)
    for (int j = 1; j <= n; ++j) anc[i][j] = anc[i - 1][anc[i - 1][j]];
  Info cur(1, 1, 0);
  for (int i = 2; i <= n; ++i) {
    Info p(cur.x, i, dist(cur.x, i));
    Info q(cur.y, i, dist(cur.y, i));
    cur = std::max(cur, std::max(p, q));
    printf("%d ", cur.z);
  }
  return 0;
}
