#include<cstdio>
#include<iostream>
#include<algorithm>
#include<cstring>
#include<cmath>
#include<vector>
//#include<set>
//#include<queue>
//#include<stack>
#define PII pair<int,int>
#define MP make_pair
using namespace std;
typedef long long LL;
typedef unsigned long long ULL;
inline LL read()
{
	LL s=0; bool w=0; char ch=getchar();
	while(ch<'0'||ch>'9'){if(ch=='-') w=1; ch=getchar();}
	while(ch>='0'&&ch<='9'){s=(s<<3)+(s<<1)+(ch^48); ch=getchar();}
	return w ? ~s+1 : s;
}

const int maxn = 400005;
const LL infl = 9223372036854775807;

struct Node
{
	int l,r;
	vector<int> son;
	vector<int> typ;
}tr[maxn];
int tot;

int n,m;
LL a[maxn];
char str[maxn];

void MakeNode(int l,int r,int fa,int x) { tr[x].l = l; tr[x].r = r; tr[fa].son.emplace_back(x); }

void Build(int l,int r,int x)
{
	//printf("%d %d~%d\n",x,l,r);
	if(l>=r) return;
	LL minn = infl;
	for(int i=l; i<=r; i++) minn = min(minn,a[i]);
	int lst = l;
	tr[x].son.emplace_back(0); tr[x].typ.emplace_back(0);
	for(int i=l; i<=r; i++)
	{
		if(a[i] == minn)
		{
			if(lst<=i-1) { MakeNode(lst,i-1,x,++tot); Build(lst,i-1,tot); tr[x].typ.emplace_back(0); }
			MakeNode(i,i,x,++tot); Build(i,i,tot); tr[x].typ.emplace_back(1); lst = i+1;
		}
		else if(i==r) { MakeNode(lst,i,x,++tot); Build(lst,i,tot); tr[x].typ.emplace_back(0); }
	}
	//printf("son of %d: ",x);
	//for(int i=1; i<(int)tr[x].son.size(); i++) printf("%d[%d] ",tr[x].son[i],tr[x].typ[i]);
	//printf("\n");
	return;
}

struct DP1
{
	LL val;
	int num;
	DP1(){}; DP1(LL v_,int n_){val = v_; num = n_;}
}f[maxn],g[maxn];
LL h[maxn];
LL sum[maxn],pre[maxn];

int hd,tl;
struct DP2
{
	LL x,y;
	int num;
	DP2(){}; DP2(LL x_,LL y_,int n_){x=x_; y=y_; num=n_;}
}que[maxn<<2];

bool Convex(DP2 x,DP2 y,DP2 z) //如果三个点构成下凸壳返回true
{
	LL dif = (z.y-y.y)*(y.x-x.x) - (z.x-y.x)*(y.y-x.y);
	return dif==0 ? z.num < y.num : dif>0; //共线的话，保证选的边数是递减的
}
bool Tangent(DP2 x,DP2 y,LL tng){return (y.y-x.y) < (y.x-x.x)*tng;}

void Pop(LL tng)
{
	while(tl>hd && Tangent(que[hd],que[hd+1],tng)) hd++;
	while(hd<tl && tng*(que[hd+1].x - que[hd].x) == que[hd+1].y-que[hd].y) hd++;
}
void Push(DP2 t)
{
	while(tl>hd && !Convex(que[tl-1],que[tl],t)) tl--;
	while(tl>=hd && que[tl].x == t.x && que[tl].y == t.y && que[tl].num < t.num) tl--;
	que[++tl] = t; return;
}

void DP(int u,LL K)
{
	if(tr[u].l > tr[u].r) {f[u] = DP1(0,0); return;}
	if(tr[u].l == tr[u].r) {f[u] = DP1(-K,1); return;}
	int sz = tr[u].son.size()-1;
	for(int i=1; i<=sz; i++) DP(tr[u].son[i],K);
	for(int i=1; i<=sz; i++)
	{
		pre[i] = pre[i-1] + tr[u].typ[i];
		sum[i] = sum[i-1] + h[tr[u].son[i]];
	}
	//printf("now at %d\n",u);
	hd = 1; tl = 0;
	Push(DP2(0,0,0));
	for(int i=1; i<=sz; i++)
	{
		Pop(pre[i]);
		g[i] = DP1(f[tr[u].son[i]].val + sum[i-1] + ((pre[i]-1)*pre[i])/2 +que[hd].y -pre[i]*que[hd].x,
				   f[tr[u].son[i]].num+que[hd].num);
		//for(int i=hd; i<=tl; i++)
		//{
			//printf("(%lld %lld %d) ",que[i].x,que[i].y,que[i].num);
		//}
		//printf("\n");
		Push(DP2(pre[i-1],g[i].val-sum[i]+((pre[i-1]+1)*pre[i-1])/2,g[i].num));
	}
	LL tmp = 0;
	f[u].val = infl; f[u].num = 0;
	//for(int i=1; i<=sz; i++) printf("(%lld,%d) ",g[i].val,g[i].num);
	//printf("\n");
	for(int i=sz; i>=1; i--)
	{
		g[i].val += tmp + ((pre[sz]-pre[i-1]-1)*(pre[sz]-pre[i-1]))/2;
		tmp += h[tr[u].son[i]];
		if(f[u].val > g[i].val || (f[u].val == g[i].val && f[u].num < g[i].num)) { f[u] = g[i]; }
	}
	//printf("f[%d] = %lld,%d\n",u,f[u].val,f[u].num);
	h[u] = sum[sz] + ((pre[sz]-1)*pre[sz])/2; return;
}

int main()
{
	//freopen("input.txt","r",stdin);
	n = read(); m = read();
	scanf("%s",str+1);
	for(int i=1; i<=n; i++) a[i+1] = a[i] + (str[i] == '(' ? 1 : -1);
	//for(int i=1; i<=n+1; i++) printf("%d ",a[i]);
	//printf("\n");
	n++;
	tr[++tot].l = 1; tr[1].r = n;
	Build(1,n,1);
	LL l = -1ll*n*n,r = 0;
	LL mid;
	while(l<r)
	{
		mid = (l+r)>>1;
		//printf("K=%lld\n",mid);
		DP(1,mid); if(f[1].val > h[1]){f[1].val=h[1]; f[1].num=0;}
		if(f[1].num >= m-1) {r = mid;}
		else l = mid+1;
	}
	DP(1,l); if(f[1].val > h[1]) {f[1].val = h[1]; f[1].num = 0;}
	printf("%lld\n",f[1].val + l*(m-1));
	//printf("%d\n",((-1)>>1));
	//fclose(stdin);
	return 0;
}