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#include <bits/stdc++.h>
using namespace std;
#define rep(i,a,n) for (int i=a;i<n;i++)
#define per(i,a,n) for (int i=n-1;i>=a;i--)
#define pb push_back
#define eb emplace_back
#define mp make_pair
#define all(x) (x).begin(),(x).end()
#define fi first
#define se second
#define SZ(x) ((int)(x).size())
typedef vector<int> VI;
typedef basic_string<int> BI;
typedef long long ll;
typedef pair<int,int> PII;
typedef double db;
mt19937 mrand(1);
const ll mod=1000000007;
int rnd(int x) { return mrand() % x;}
ll powmod(ll a,ll b) {ll res=1;a%=mod; assert(b>=0); for(;b;b>>=1){if(b&1)res=res*a%mod;a=a*a%mod;}return res;}
//ll gcd(ll a,ll b) { return b?gcd(b,a%b):a;}
// head

const int N=101000;
int n,k,m,ps[N],pre[N],nxt[N],lab[N],qs[N];
int bg[N],ed[N];
ll sig[N],sm[N],ls[N],rs[N];
char s[N];

const int LOGN=19;
template<class T>
struct STTable {
	int n;
	T f[LOGN+1][N];
	void init(vector<T> a) { // 0-based
		n=SZ(a);
		assert(n < (1<<LOGN) && n < N);
		rep(i,0,n) f[0][i]=a[i];
		rep(j,1,LOGN) for (int i=0;i+(1<<j)-1<n;i++) {
			f[j][i]=min(f[j-1][i],f[j-1][i+(1<<(j-1))]);
		}
	}
	T query(int l,int r) {
		assert(l<=r);
        int len=31-__builtin_clz(r-l+1);
        return min(f[len][l],f[len][r-(1<<len)+1]);
	}
};
ll f[N];
int opt[N];
STTable<PII> f1,f2;
int main() {
	scanf("%d%d",&n,&k);
	scanf("%s",s);
	for (int i=1;i<=n;i++) {
		if (s[i-1]=='(') ps[i]=ps[i-1]+1;
		else ps[i]=ps[i-1]-1;
	}
	map<int,int> pos;
	auto getsmseg=[&](int l,int r) {
		assert(lab[l]==lab[r]&&l<=r);
		return sm[r]-sm[l]+(ll)(qs[r]-qs[l])*(qs[r]-qs[l]+1)/2;
	};
	for (int i=0;i<=n;i++) {
		if (!pos.count(ps[i])) pre[i]=-1;
		else {
			int l=pos[ps[i]];
			if (!pos.count(ps[i]-1)||pos[ps[i]-1]<l)
				pre[i]=l;
			else pre[i]=-1;
		}
		//printf("%d %d\n",i,pre[i]);
		pos[ps[i]]=i;
		if (pre[i]==-1) {
			lab[i]=++m;
			qs[i]=0;
			sig[i]=0;
			bg[i]=i;
		} else {
			lab[i]=lab[pre[i]];
			bg[i]=bg[pre[i]];
			qs[i]=qs[pre[i]]+1;
			sig[i]=getsmseg(pre[i]+1,i-1);
			sm[i]=sm[pre[i]]+sig[i];
		}
	}
	rep(i,0,n+1) nxt[i]=-1;
	rep(i,0,n+1) if (pre[i]!=-1) nxt[pre[i]]=i;
	per(i,0,n+1) {
		if (nxt[i]==-1) ed[i]=i;
		else ed[i]=ed[nxt[i]];
	}
	per(i,0,n+1) rs[i]=rs[ed[i]+1]+getsmseg(i,ed[i]);
	rep(i,0,n+1) ls[i]=(bg[i]?ls[bg[i]-1]:0)+getsmseg(bg[i],i);
	vector<PII> mv(n+1);
	rep(i,0,n+1) mv[i]=mp(ps[i],i);
	f1.init(mv);
	rep(i,0,n+1) mv[i]=mp(ps[i],-i);
	f2.init(mv);
	auto weight=[&](int l,int r) {
		if (l>r) return 0ll;
		//assert(l<=r);
		PII minu=f1.query(l-1,r),minv=f2.query(l-1,r);
		int u=minu.se,v=-minv.se;
		return getsmseg(u,v)+rs[l-1]-rs[u]+ls[r]-ls[v];
	};
	auto solve=[&](ll c,int ty=0){
		static int vo[N],po[N];
		int p=1,q=0;
		f[0]=0;
		auto val=[&](int j,int i) { //let val(j,i) = f[j]+w(j,i)
			return f[j]+weight(j+1,i)+c;
		};
		auto check=[&](int p,int q,int r) {
			return (ty==0)?val(p,r)<=val(q,r):val(p,r)<val(q,r);
		};
		rep(i,0,n){
			while(p<=q && check(i,vo[q],po[q]))q--;//
			if(p>q)po[++q]=i+1,vo[q]=i;
			else{
				int l=po[q],r=n;
				while(l<=r){
					int mid=(l+r)>>1;
					if(check(i,vo[q],mid))r=mid-1;//
					else l=mid+1;
				}
				if(l<=n) po[++q]=l,vo[q]=i;
			}
			while(p<q && i+1>=po[p+1])p++;
			f[i+1]=val(vo[p],i+1);
			opt[i+1]=vo[p];
		}
		int x=n,cnt=0;
		while (x!=0) x=opt[x],cnt++;
		return cnt;
	};
	ll pl=0,pr=(ll)n*n;
	while (pl+1<pr) {
		ll md=(pl+pr)>>1;
		int w=solve(md,0);
		if (w>=k) pl=md; else pr=md;
	}
	solve(pl,0);
	printf("%lld\n",f[n]-k*pl);
}