1481_F. AB Tree

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.

<image>

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:

<image>

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.

Example input:
9 3
1 2 2 4 4 4 3 1


Example output:

4
aabbbbbba


Tags: dp
greedy
trees
Difficulty rating: 3100

Solution:
#include <iostream>
#include <cstdio>
#include <algorithm>
#include <string.h>
#include <vector>
using namespace std;
#define endl '\n'
#define ll long long
#define pi pair<int, int>
#define f first
#define s second

const int mxn = 100001;
int n, m, k;
int d[mxn], f[mxn], dp[mxn], p[mxn];
vector<int> g[mxn], a[mxn], v[mxn];

void dfs(int c){
	m = max(m, d[c]);
	a[d[c]].push_back(c);
	for(int i : g[c]) d[i] = d[c] + 1, dfs(i);
}

int main(){
	ios::sync_with_stdio(0);
	cin.tie(0);
	
	cin >> n >> k;
	
	for(int i = 1; i < n; i++){
		int x;
		cin >> x;
		g[--x].push_back(i);
	}
	
	dfs(0);
	
	m++;
	for(int i = 0; i < m; i++) v[a[i].size()].push_back(i);
	
	memset(dp, -1, sizeof(dp));
	dp[0] = -2;
	for(int i = 0; i <= n; i++) if(!v[i].empty())
	for(int j = n; ~j; j--) if(~dp[j])
	for(int l = 0; l < v[i].size() && !~dp[j + (l + 1) * i]; l++){
		dp[j + (l + 1) * i] = v[i][l];
	}
	
	if(~dp[k]){
		cout << m << endl;
		for(int i = k; i; i -= a[dp[i]].size())
		for(int j : a[dp[i]]){
			f[j] = 1;
		}
	}else{
		cout << (m + 1) << endl;
		int w = 0, x = 0, y = k, z = n - k;
		for(int i = 0, j = 1; i < m; i++, x += j){
			if(a[i].size() > max(y, z)){
				j = 0;
				continue;
			} 
			for(int l : a[i]) f[l] = (y > z);
			(y > z ? y : z) -= a[i].size();
		}
		for(int i : a[x]){
			if(!g[i].empty()) f[i] = (y > z), w++;
		}
		(y > z ? y : z) -= w;
		for(int i : a[x]){
			if(g[i].empty()) f[i] = y > z, (y > z ? y : z)--;
		}
	}
	
	for(int i = 0; i < n; i++) cout << (f[i] ? 'a' : 'b');
	
	return 0;
}