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.

<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.

Solution (in CPP):

Solution:
#include<bits/stdc++.h>
using namespace std;
typedef long long ll;

const int N=1e5+10;
const int K=600;

vector<int>e[N],v1[N],v2[N];
int dep[N],siz[N];
int flag[N];
int d;
int a[N];
int ans[N];

bitset<N>dp[K];

void dfs(int u,int fa){ 
	siz[u]=1;
	dep[u]=dep[fa]+1;
	d=max(d,dep[u]);
	v1[dep[u]].push_back(u);
	for(auto v: e[u]){
		dfs(v,u);
		siz[u]+=siz[v];
	}
}
int main()
{
	int n,x;
	scanf("%d%d",&n,&x);
	for(int i=2;i<=n;i++){
		int p;
		scanf("%d",&p);
		e[p].push_back(i);
	}
	dfs(1,0);
	for(int i=1;i<=d;i++){
		v2[v1[i].size()].push_back(i);
	}
	int m=0;
	for(int i=1;i<=n;i++){
		if(v2[i].size()){
			a[++m]=i;
		}
	}
	dp[0][0]=1;
	for(int i=1;i<=m;i++){
		dp[i]=dp[i-1];
		int u=a[i];
		int sz=v2[u].size();
		for(int j=1;j<=sz;j<<=1){
			dp[i]|=(dp[i]<<(u*j));
			sz-=j;
		}
		if(sz) dp[i]|=(dp[i]<<(u*sz));
	}
	if(dp[m][x]){
		for(int i=m;i>=1;i--){
			int u=a[i];
			for(auto v: v2[u]){
				if(dp[i-1][x])
					break;
				x-=u;
				flag[v]=1;
			}
		}
		printf("%d\n",d);
	}
	else{
		printf("%d\n",d+1);
		int y=n-x;
		for(int i=1;i<=d;i++){
			int sz=v1[i].size();
			if(x>=sz){
				flag[i]=1;
				x-=sz;
			}
			else if(y>=sz){
				y-=sz;
			}
			else{
				flag[i]=-1;
				bool f=(x>y);
				for(auto v: v1[i]){
					if(siz[v]>1){
						if(f) ans[v]=1,x--;
						else y--;
					}
				}
				for(auto v: v1[i]){
					if(siz[v]==1){
						if(f&&x>0) ans[v]=1,x--;
						else if(f&&x==0) y--;
						else if(!f&&y>0) y--;
						else if(!f&&y==0) ans[v]=1,x--;
					}
				}
			}
		}
	}
	for(int i=1;i<=n;i++){
		if(flag[dep[i]]==1||(flag[dep[i]]==-1&&ans[i])){
			printf("a");
		}
		else printf("b");
	}
	printf("\n");
	return 0;
}