Problem: Kilani and Abd are neighbors for 3000 years, but then the day came and Kilani decided to move to another house. As a farewell gift, Kilani is going to challenge Abd with a problem written by their other neighbor with the same name Abd.

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The problem is:

You are given a connected tree rooted at node 1.

You should assign a character a or b to every node in the tree so that the total number of a's is equal to x and the total number of b's is equal to n - x.

Let's define a string for each node v of the tree as follows: 

  * if v is root then the string is just one character assigned to v: 
  * otherwise, let's take a string defined for the v's parent p_v and add to the end of it a character assigned to v. 



You should assign every node a character in a way that minimizes the number of distinct strings among the strings of all nodes.

Input

The first line contains two integers n and x (1 ≤ n ≤ 10^5; 0 ≤ x ≤ n) — the number of vertices in the tree the number of a's.

The second line contains n - 1 integers p_2, p_3, ..., p_{n} (1 ≤ p_i ≤ n; p_i ≠ i), where p_i is the parent of node i.

It is guaranteed that the input describes a connected tree.

Output

In the first line, print the minimum possible total number of distinct strings.

In the second line, print n characters, where all characters are either a or b and the i-th character is the character assigned to the i-th node.

Make sure that the total number of a's is equal to x and the total number of b's is equal to n - x.

If there is more than one answer you can print any of them.

Example

Input


9 3
1 2 2 4 4 4 3 1


Output


4
aabbbbbba

Note

The tree from the sample is shown below:

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The tree after assigning characters to every node (according to the output) is the following:

<image>

Strings for all nodes are the following: 

  * string of node 1 is: a
  * string of node 2 is: aa
  * string of node 3 is: aab
  * string of node 4 is: aab
  * string of node 5 is: aabb
  * string of node 6 is: aabb
  * string of node 7 is: aabb
  * string of node 8 is: aabb
  * string of node 9 is: aa



The set of unique strings is \{a, aa, aab, aabb\}, so the number of distinct strings is 4.

Solution (in CPP):

Solution:
#include <bits/stdc++.h>
const int N = 1e5 + 5;
int n, k, max, a[N], p[N], tot[N], cnt[N], f[N], g[N], fre[N], mx[N];
std::vector<int> nodes[N], adj[N];
bool leaf[N], ans[N];
void dfs(int u, int dep) {
  ++tot[dep], max = std::max(max, dep + 1), nodes[dep].push_back(u);
  for (auto v : adj[u]) dfs(v, dep + 1);
}
int main() {
  scanf("%d%d", &n, &k);
  memset(leaf, true, sizeof leaf);
  for (int i = 2; i <= n; ++i) 
    scanf("%d", &p[i]), adj[p[i]].push_back(i), leaf[p[i]] = false;
  dfs(1, 0), memset(f, -1, sizeof(f)), f[0] = 0;
  for (int i = 0; i < max; ++i) ++fre[tot[i]];
  for (int i = 1; i <= n; ++i) if (fre[i]) {
    memset(mx, 0xcf, sizeof(mx));
    for (int j = 0; j <= n; ++j) {
      int rem = j % i, k = mx[rem];    
      if (~f[j]) mx[rem] = j;
      else if (k + i * fre[i] >= j)
        f[j] = i, g[j] = (j - k) / i;
    }
  }
  if (~f[k]) {
    printf("%d\n", max);
    for (int i = k; i; cnt[f[i]] += g[i], i -= f[i] * g[i]);
    for (int i = 0; i < max; ++i) if (cnt[tot[i]]) {
      --cnt[tot[i]]; for (auto u : nodes[i]) ans[u] = 1;
    }
  } else {
    printf("%d\n", max + 1);
    for (int i = 0, m = n; i < max; m -= tot[i++]) {
      if (k >= tot[i]) for (auto u : nodes[i]) ans[u] = 1, --k;
      else if (m - k < tot[i]) {
        std::sort(nodes[i].begin(), nodes[i].end(), [&](int i, int j) {return leaf[i] < leaf[j];});
        if (k <= m - k) std::reverse(nodes[i].begin(), nodes[i].end());
        for (auto it = nodes[i].begin(); k--; ans[*it++] = 1);
      }
    }
  }
  for (int i = 1; i <= n; ++i) printf("%c", ans[i] ? 'a' : 'b');
  return 0;
}