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#include <cstdio>
#include <ctime>
#include <deque>
#include <cstring>
#include <algorithm>

const int N = 100005;
const int K = 200;
int n, k, pr[N], size = 1, a[N], licz[N / K + 1][N], licz1[N];
unsigned duz[N / K + 1][N];
char s[N];
std::pair<long long, int> d[N];

void init() {
	for (int i = 0; (i + 1) * K <= n + 1; ++i) {
		if (i) for (int j = 0; j <= n; ++j) licz[i][j] = licz[i - 1][j];
		for (int j = i * K; j < i * K + K; ++j) ++licz[i][a[j]];
	}
	for (int i = 0; (i + 1) * K <= n + 1; ++i) {
		long long s = 0;
		for (int j = i * K + K - 1; j >= 0; --j) {
			s += licz1[a[j]]++;
			duz[i][j] = s;
		}
		for (int i = 0; i <= n; ++i) licz1[i] = 0;
	}
}

long long f(int l, int r) {
	long long ans = 0;
	if (r - l + 1 <= 2 * K) {
		for (int i = l; i <= r; ++i) ans += licz1[a[i]]++;
		for (int i = l; i <= r; ++i) licz1[a[i]] = 0;
	} else {
		int bl = l / K, br = r / K;
		if (bl * K != l) ++bl;
		if (br * K + K - 1 != r) --br;
		ans = duz[br][l];
		for (int i = l; i < bl * K; ++i) ++licz1[a[i]];
		for (int i = br * K + K; i <= r; ++i) {
			ans += licz1[a[i]]++;
			ans += licz[br][a[i]] - (bl ? licz[bl - 1][a[i]] : 0);
		}
		for (int i = l; i < bl * K; ++i) licz1[a[i]] = 0;
		for (int i = br * K + K; i <= r; ++i) licz1[a[i]] = 0;
	}
	return ans;
}

int licz2[N];
long long f1(int l, int r) {
	long long ans = 0;
	for (int i = l; i <= r; ++i) ans += licz2[a[i]]++;
	for (int i = l; i <= r; ++i) licz2[a[i]] = 0;
	return ans;
}

int call = 0;
int lepej(int x, int y) {
	++call;
	int b;
	int L = y / K, R = (n + 1) / K;
	while (L < R) {
		int M = L + R >> 1;
		if (std::make_pair(d[x].first + duz[M][x], d[x].second) >= std::make_pair(d[y].first + duz[M][y], d[y].second)) R = M;
		else L = M + 1;
	}
	b = R;
	int bx = x / K;
	if (bx * K != x) ++bx;
	int by = y / K;
	if (by * K != y) ++by;
	if (b > y / K) {
		for (int i = x; i < bx * K; ++i) ++licz1[a[i]];
		for (int i = y; i < by * K; ++i) --licz1[a[i]];
		long long ws = d[x].first - d[y].first + duz[b - 1][x] - duz[b - 1][y];
		int p;
		for (p = b * K; p <= n; ++p) {
			ws += licz1[a[p]];
			ws += -(bx ? licz[bx - 1][a[p]] : 0) + (by ? licz[by - 1][a[p]] : 0);
			if (ws > 0 || !ws && d[x].second >= d[y].second) {
				for (int i = x; i < bx * K; ++i) licz1[a[i]] = 0;
				for (int i = y; i < by * K; ++i) licz1[a[i]] = 0;
				return p;
			}
		}
		for (int i = x; i < bx * K; ++i) licz1[a[i]] = 0;
		for (int i = y; i < by * K; ++i) licz1[a[i]] = 0;
		return p;
	}
	if (y - x + 1 <= 2 * K) {
		long long ws = d[x].first - d[y].first;
		for (int i = x; i <= y; ++i) ws += licz1[a[i]]++;
		--licz1[a[y]];
		int p;
		for (p = y + 1; p <= n; ++p) {
			ws += licz1[a[p]];
			if (ws > 0 || !ws && d[x].second >= d[y].second) {
				for (int i = x; i <= y; ++i) licz1[a[i]] = 0;
				return p;
			}
		}
		for (int i = x; i <= y; ++i) licz1[a[i]] = 0;
		return p;
	}
	if (b * K + K - 1 == y) return y + 1;
	long long ws = d[x].first - d[y].first;
	for (int i = x; i < bx * K; ++i) ++licz1[a[i]];
	ws += duz[b - 1][x];
	for (int i = b * K; i <= y; ++i) {
		ws += licz1[a[i]]++;
		ws += licz[b - 1][a[i]] - (bx ? licz[bx - 1][a[i]] : 0);
	}
	--licz1[a[y]];
	int p;
	for (p = y + 1; p <= n; ++p) {
		ws += licz1[a[p]];
		ws += licz[b - 1][a[p]] - (bx ? licz[bx - 1][a[p]] : 0);
		if (ws > 0 || !ws && d[x].second >= d[y].second) {
			for (int i = x; i < bx * K; ++i) licz1[a[i]] = 0;
			for (int i = b * K; i <= y; ++i) licz1[a[i]] = 0;
			return p;
		}
	}
	for (int i = x; i < bx * K; ++i) licz1[a[i]] = 0;
	for (int i = b * K; i <= y; ++i) licz1[a[i]] = 0;
	return n + 1;
}

void run(long long C) {
	d[0] = std::make_pair(0, 0);
	std::deque<std::pair<int, int> > dq;
	dq.push_back(std::make_pair(0, 0));
	for (int i = 1; i <= n; ++i) {
		while (dq.size() > 1 && dq[1].second <= i) dq.pop_front();
		d[i] = std::make_pair(d[dq[0].first].first + C + f(dq[0].first, i), d[dq[0].first].second + 1);
		while (!dq.empty()) {
			int l = lepej(dq.back().first, i);
			if (dq.back().second >= l) dq.pop_back();
			else {
				dq.push_back(std::make_pair(i, l));
				break;
			}
		}
		if (dq.empty()) dq.push_back(std::make_pair(i, 0));
	}
	call = 0;
}

int main() {
	//n = 100000;
	//k = 500;
	scanf("%d%d", &n, &k);
	//for (int i = 0; i < n; ++i) s[i] = i & 1 ? ')' : '(';
	scanf("%s", s);
	//for (int i = 0; i < n; ++i) s[i] = rand() & 1 ? '(' : ')';
	int cur = 0;
	pr[0] = -1;
	for (int i = 0; i <= n; ++i) {
		a[i] = cur;
		if (i < n) {
			if (s[i] == '(') {
				pr[size] = cur;
				cur = size++;
			} else {
				if (pr[cur] == -1) {
					pr[size] = -1;
					pr[cur] = size++;
				}
				cur = pr[cur];
			}
		}
	}
	init();
	long long l = 0, r = (long long)N * N;
	while (l < r) {
		long long m = l + r >> 1;
		run(m);
		if (d[n].second <= k) r = m;
		else l = m + 1;
	}
	run(r);
	printf("%lld\n", d[n].first - (long long)k * r);
	//fprintf(stderr, "%lf\n", (double)clock() / CLOCKS_PER_SEC);
	return 0;
}