#include<bits/stdc++.h>
using namespace std;
int s[500100],is[500100][20];
#define ll long long
#define pb push_back
int cz(int x,int y){if(s[x]<s[y]||s[x]==s[y]&&x<y)return x;return y;}
int lg[501000];
int as(int l,int r){
	int k=lg[r-l+1];
	return cz(is[l][k],is[r-(1<<k)+1][k]);
}
ll sz[500100];
ll C(ll x){return x*(x-1)/2;}
char c[501000];int cn;
vector<int>g[501000];
int build(int l,int r){
	if(l>r)return 0;
	int u=++cn;
	int m=as(l,r);
	g[u].pb(build(l,m-1));
	while(m<r){
		int v=as(m+1,r);if(s[m]!=s[v])break;
		g[u].pb(build(m+1,v-1)),m=v;
	}
	g[u].pb(build(m+1,r));
	for(int v:g[u])sz[u]+=sz[v];
	sz[u]+=C(((int)g[u].size())-1);
	return u;
}
int M;
#define mp make_pair
#define fi first
#define se second
#define pi pair<ll,ll>
const ll I=1e18;
pi dp[501000],gd[500100],f[500100];
void ad(pi &a,pi b){if(a.fi>b.fi||a.fi==b.fi&&a.se<b.se)a=b;}
pi oo(pi a,ll b){return mp(a.fi+b,a.se);}
ll su[501000];
void dfs(int x){
	if(!x)return;
	for(int v:g[x])dfs(v);
	int n=g[x].size();
	for(int i=1;i<=n;i++)su[i]=su[i-1]+sz[g[x][i-1]],gd[i]=dp[g[x][i-1]];
	dp[x]=mp(I,0);
	for(int i=1;i<=n;i++){
		f[i]=mp(C(i-1),0);
		for(int j=1;j<i;j++)ad(f[i],oo(f[j],-su[j]+C(i-j)));
		f[i].fi+=su[i-1]+gd[i].fi,f[i].se+=gd[i].se;
		ad(dp[x],oo(f[i],C(n-i)+su[n]-su[i]));
	}
}
bool chk(ll mid){
	dp[0]=mp(mid,1),dfs(1);
	ad(dp[1],mp(sz[1],0));
	return dp[1].se>=M;
}
int n;
int main(){
	scanf("%d%d%s",&n,&M,c+1);
	M--;
	lg[0]=-1;
	for(int i=1;i<=n;i++)s[i+1]=s[i]+((c[i]=='(')?1:-1),lg[i]=lg[i>>1]+1;
	n++;
	for(int i=1;i<=n;i++)is[i][0]=i;
	for(int j=1;j<20;j++)
		for(int i=1;i+(1<<j)-1<=n;i++)
		is[i][j]=cz(is[i][j-1],is[i+(1<<(j-1))][j-1]);
	build(1,n);
	ll l=0,r=1e12,ae=-1;
	while(l<=r){
		ll mi=(l+r)/2;
		if(chk(mi))ae=mi,l=mi+1;
		else r=mi-1;
	}
	chk(ae);
	printf("%lld\n",dp[1].fi-ae*M);
}