#include <stdio.h>
#include <algorithm>
#include <string.h>
#include <cctype>
#include <vector>
#include <queue>
#include <bitset>
#include <cmath>
#define vi vector<int>
#define vl vector<long long>
#define pb push_back
#define mp make_pair
#define st first
#define nd second
using namespace std;
typedef long long ll;
typedef pair <int, int> Pii;
const int INF=0x3f3f3f3f;
const int cp=998244353;
inline int mod(int x){if(x>=cp) x-=cp;if(x<0) x+=cp;return x;}
inline void plust(int &x, int y){x=mod(x+y);return ;}
inline void minut(int &x, int y){x=mod(x-y);return ;}
inline int read(){
	char ch=getchar();int x=0, f=1;
	while(!isdigit(ch)){if(ch=='-') f=-1; ch=getchar();}
	while(isdigit(ch)){x=(x<<3)+(x<<1)+ch-'0';ch=getchar();}
	return x*f;
}
inline void write(int x){
    if(x<0) putchar('-'), x=-x;
    if(x>9) write(x/10);
    putchar(x%10+'0');
}
inline int ksm(int a, int b=cp-2){
	int ret=1;
	for(; b; b>>=1, a=1ll*a*a%cp)
		if(b&1) ret=1ll*ret*a%cp;
	return ret;
}
const int N=1e5+5;
const int M=N<<1;
int n, K, sum[N];char s[N];
namespace CarTree{//Cartesian Tree
	int cnt, pos[N], dfn[M], ST[20][N], ET[20][M], val[M];
	int fa[M][20], son[M], mxd[M], dep[M], top[M], pc[M], sc[M];
	bool tp[M];vi G[M], U[M], D[M];
	ll pre[M], suf[M], C2[N], siz[M], ep[M], es[M], S[M];
	//若 x 是圆点，则 val 表示 x 有多少个方点儿子
	//若 x 是方点，则 val 表示 x 是父亲的第几个方点儿子
	int MN(int x, int y){return sum[x]==sum[y]?(x<y?x:y):(sum[x]<sum[y]?x:y);}
	int qry(int x, int y){
		if(x>y) return -1;int k=__lg(y-x+1);
		return MN(ST[k][x], ST[k][y-(1<<k)+1]);
	}
	int MX(int x, int y){return dep[x]<dep[y]?x:y;}
	int LCA(int x, int y){
		if(x==y) return x;x=dfn[x], y=dfn[y];if(x>y) swap(x, y);
		int k=__lg(y-x);return fa[MX(ET[k][x+1], ET[k][y-(1<<k)+1])][0];
	}
	void build(int k, int l, int r){
		if(l>r) return ;
		int x=qry(l, r), v=sum[x];
		// printf("[%d %d %d]\n", l, r, x);
		for(; x>=0&&sum[x]==v; l=x+1, x=qry(x+1, r)){
			if(l<x) G[k].pb(++cnt), build(cnt, l, x-1);
			G[k].pb(++cnt), tp[cnt]=1, val[cnt]=++val[k], pos[x]=cnt;
		}
		if(l<=r) G[k].pb(++cnt), tp[cnt]=0, build(cnt, l, r);
	}
	void dfs(int x){
		mxd[x]=dep[x];ET[0][dfn[x]=++dfn[0]]=x;if(!tp[x]) siz[x]=C2[val[x]];
		for(int i=1; i<20; ++i) fa[x][i]=fa[fa[x][i-1]][i-1];
		for(auto v:G[x]){
	//int c=0;suf[v]=suf[x]+C2[val[x]-c+1];if(tp[v]) ++c;pre[v]=pre[x]+C2[c];
			dep[v]=dep[x]+1, fa[v][0]=x;dfs(v);siz[x]+=siz[v];
			mxd[x]=max(mxd[v], mxd[x]);if(mxd[v]>mxd[son[x]]) son[x]=v;
		}
	}
	void dfs(int x, int rt){
		// printf(">>>%d %d\n", x, rt);
		if(x==rt) for(int i=0, p=x; i<=mxd[x]-dep[x]; ++i, p=fa[p][0]) U[x].pb(p);
		D[top[x]=rt].pb(x);if(son[x]) dfs(son[x], rt);
		for(auto v:G[x]) if(!top[v]) dfs(v, v);
	}
	void calc(int x){
		int sz=G[x].size()-1;
		ll pres=0;for(int i=0, c=0, v; i<=sz; ++i)
			c+=tp[v=G[x][i]], pre[v]=pre[x]+C2[c]+pres, 
			pc[v]=c, ep[v]=pres, pres+=siz[v];
		ll sufs=0;for(int i=sz, c=0, v; i>=0; --i)
			c+=tp[v=G[x][i]], suf[v]=suf[x]+C2[c]+sufs,
			sc[v]=c, es[v]=sufs, sufs+=siz[v];
		ll s=0;for(auto v:G[x]) S[v]=(s+=siz[v]), calc(v);
	}
	int kLA(int x, int k){
		if(!k) return x;int hb=__lg(k);
		x=fa[x][hb];k-=(1<<hb)+dep[x]-dep[top[x]];
		// printf("ask %d to %d [%d]\n", x, top[x], k);
		return (k>=0)?U[top[x]][k]:D[top[x]][-k];
	}
	ll query(int l, int r){
		if(l+1>=r) return 0;
		// printf("ask [%d %d] ", l, r);
		l=pos[l], r=pos[r];int x=LCA(l, r);
		ll p=suf[l]-suf[x]+pre[r]-pre[x];
		int lk=kLA(l, dep[l]-dep[x]-1), rk=kLA(r, dep[r]-dep[x]-1);
		// printf("%d(%d) %d(%d) to %d %lld+%lld\n", 
		// l, lk, r, rk, x, suf[l], pre[r]);
		p+=C2[(pc[rk]-pc[lk]+tp[rk]+sc[lk]-sc[rk]+tp[lk])/2]-C2[sc[lk]]-C2[pc[rk]];
		p+=(S[rk]-S[lk]-siz[rk])-es[lk]-ep[rk];
		// printf("find %lld\n", p);
		return p;
	}
	void init(){
		for(int i=1; i<=n; ++i) 
			sum[i]=sum[i-1]+(s[i]=='('?1:-1), ST[0][i]=i, C2[i]=1ll*i*(i-1)/2;
		C2[n+1]=1ll*n*(n+1)/2, sum[0]=ST[0][0]=0;
		// for(int i=0; i<=n; ++i) printf("%d", sum[i]);puts("");
		for(int i=1; i<=__lg(n+1); ++i)
			for(int x=0; x+(1<<i)-1<=n; ++x)
				ST[i][x]=MN(ST[i-1][x], ST[i-1][x+(1<<i-1)]);
		build(cnt=1, 0, n);
		dfs(dep[1]=1);dfs(1, 1);calc(1);
		// for(int i=1; i<=cnt; ++i) printf("%lld ", suf[i]);puts("");
		for(int i=1; i<=__lg(cnt); ++i)
			for(int x=1; x+(1<<i)-1<=cnt; ++x)
				ET[i][x]=MX(ET[i-1][x], ET[i-1][x+(1<<i-1)]);
	}
}
ll f[N];
int h, t, g[N];
struct node{int l, r, pos;}q[N];
ll DP(int l, int r){return f[l]+CarTree :: query(l, r);}
bool cmp(int a, int b, int r){
	ll fa=DP(a, r), fb=DP(b, r);
	return fa>fb||(fa==fb&&g[a]>=g[b]); 
}
int find(int l, int r, int p, int q){
	int res=r;
	while(l<=r){
		int mid=(l+r)/2;
		if(cmp(p, q, mid)) r=mid-1, res=mid;
		else l=mid+1;
	}
	return res;
}
int check(ll C){
	q[h=t=1]=(node){1, n, 0};
	for(int i=1; i<=n; ++i){
		while(h<=t&&q[h].r<i) ++h;
		f[i]=DP(q[h].pos, i)-C, g[i]=g[q[h].pos]+1;
		if(cmp(q[t].pos, i, n)){
			int L=n;
			while(h<=t&&cmp(q[t].pos, i, q[t].l)) L=q[t].l, --t;
			if(h<=t&&cmp(q[t].pos, i, q[t].r)) L=find(q[t].l, q[t].r, q[t].pos, i);
			if(L<=q[t].r) q[t].r=L-1;q[++t]=(node){L, n, i};
		}
	}
	return g[n];
}
signed main(){
	n=read(), K=read();
	scanf("%s", s+1);CarTree :: init();
	ll L=-2e10, R=2e10, ans=0;//斜率单增
	while(L<=R){
		ll mid=L+R>>1;
		if(check(mid)<=K) L=mid+1, ans=f[n]+mid*K;
		else R=mid-1;
	}
	printf("%lld\n", ans);
	return 0;
}