#include <bits/stdc++.h>
#define ll long long
#define ull unsigned ll
#define pir pair <ll, ll>
#define mkp make_pair
#define fi first
#define se second
#define pb push_back
using namespace std;
const ll maxn = 1e5 + 10, mod = 1e9 + 7, inf = 1e14;
ll n, k, stk[maxn], top, cnt, ggc, lpos[maxn], rpos[maxn];
char s[maxn];
vector <ll> son[maxn];
pir dp[maxn], f[maxn];
ll gain, q[maxn], l, r, sum[maxn], usum[maxn];
ll getx(ll i) {return i;}
ll gety(ll i) {return i * (i + 2) + 2 * (dp[i].fi - sum[i]);}
ll dx(ll i, ll j) {return getx(j) - getx(i);}
ll dy(ll i, ll j) {return gety(j) - gety(i);}
void dfs(ll u) {
	if(son[u].empty()) {
		usum[u] = 0;
		if(lpos[u] > 0 && rpos[u] <= n) f[u] = mkp(gain, 1);
		else f[u] = mkp(inf, 0);
		return;
	}
//	printf("dfs %lld\n", u);
	for(ll v: son[u]) dfs(v);
	l = 1, r = 0; ll cc = 0;
	for(ll i = 0; i < son[u].size(); i++) {
		ll v = son[u][i]; sum[i] = usum[v];
		if(i) sum[i] += sum[i - 1];
		dp[i].fi = f[v].fi + (cc + 1) * cc / 2 + (i? sum[i - 1] : 0);
		dp[i].se = f[v].se;
		cc += (lpos[v] > 0 && rpos[v] <= n);
		ll tmp = 1e18;
		while(l < r && dy(q[l], q[l + 1]) <= (__int128) dx(q[l], q[l + 1]) * 2 * i) {
			if(dy(q[l], q[l + 1]) == (__int128) dx(q[l], q[l + 1]) * 2 * i)
				tmp = min(tmp, dp[q[l]].se);
			++l;
		} tmp = min(tmp, dp[q[l]].se);
		if(i) { pir w; ll j = q[l];
			w.fi = dp[j].fi + sum[i - 1] - sum[j] + f[v].fi + 
			       (i - j) * (i - j - 1) / 2;
			w.se = tmp + f[v].se;
			dp[i] = min(dp[i], w);
		}
		while(l < r && (__int128) dy(q[r - 1], q[r]) * dx(q[r], i)
		> (__int128) dy(q[r], i) * dx(q[r - 1], q[r])) --r;
		q[++r] = i;
	} usum[u] = sum[son[u].size() - 1], f[u].fi = 1e18;
	for(ll i = 0; i < son[u].size(); i++) {
		pir w; ll gg = son[u].size() - i - 1;
		if(rpos[son[u][son[u].size() - 1]] > n) --gg;
		w.fi = dp[i].fi + usum[u] - sum[i]
		       + gg * (gg + 1) / 2;
		w.se = dp[i].se;
		f[u] = min(f[u], w);
	}
	usum[u] += cc * (cc + 1) / 2;
}
int main() {
//	freopen("p.in", "r", stdin);
//	freopen("p.out", "w", stdout);
	scanf("%lld%lld%s", &n, &k, s + 1);
	for(ll i = 1; i <= n; i++)
		if(s[i] == '(') ++cnt;
		else --cnt, ggc = min(ggc, cnt);
	cnt = 0, stk[top = 1] = cnt = 1;
	while(ggc++) {
		son[stk[top]].pb(++cnt);
		lpos[stk[++top] = cnt] = 0;
	}
	for(ll i = 1; i <= n; i++)
		if(s[i] == '(') {
			son[stk[top]].pb(++cnt);
			lpos[stk[++top] = cnt] = i;
		}
		else rpos[stk[top--]] = i;
	while(top > 1) rpos[stk[top--]] = n + 1;
	ll lo = 0, hi = 1e10, ans;
	while(lo <= hi) {
		gain = lo + hi >> 1, dfs(1);
		pir tmp = min(f[1], mkp(usum[1], 0ll));
		if(tmp.se < k) {
			ans = tmp.fi - gain * (k - 1);
			hi = gain - 1;
		} else lo = gain + 1;
	}
	printf("%lld", ans);
	return 0;
}