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#include <bits/stdc++.h>

#define ALL(x) (x).begin(), (x).end()
#define SZ(x) ((int)(x).size())
#define st first
#define nd second

using namespace std;
 
#define sim template < class c
#define ris return * this
#define dor > debug & operator <<
#define eni(x) sim > typename \
	enable_if<sizeof dud<c>(0) x 1, debug&>::type operator<<(c i) {
sim > struct rge { c b, e; };
sim > rge<c> range(c i, c j) { return rge<c>{i, j}; }
sim > auto dud(c* x) -> decltype(cerr << *x, 0);
sim > char dud(...);
struct debug {
#ifdef LOCAL
~debug() { cerr << endl; }
eni(!=) cerr << boolalpha << i; ris; }
eni(==) ris << range(begin(i), end(i)); }
sim, class b dor(pair < b, c > d) {
	ris << "(" << d.first << ", " << d.second << ")";
}
sim dor(rge<c> d) {
	*this << "[";
	for (auto it = d.b; it != d.e; ++it)
	*this << ", " + 2 * (it == d.b) << *it;
	ris << "]";
}
#else
sim dor(const c&) { ris; }
#endif
};
#define imie(...) " [" << #__VA_ARGS__ ": " << (__VA_ARGS__) << "] "
 
using ll = long long;
using pii = pair<int,int>;
using pll = pair<ll,ll>;
using vi = vector<int>;
using vll = vector<ll>;

template <typename T> void mini(T &a, T b) { a = min(a, b); }


const ll kInf = 1e18;

inline ll DivFloor(ll a, ll b) {
	assert(b > 0);
	ll res = a / b;
	if (res * b > a) { --res; }
	return res;
}

struct DecSeq {
	ll cur_delta = 0;
	int num_elems = 0;
	int cur_delta_ptr = 0;
	
	struct LineInfo {
		ll start;
		pll line;
		
		bool operator<(const LineInfo &L) const {
			return start < L.start;
		}
		
		ll Eval(ll x) const {
			return line.st * x + line.nd;
		}
	};
	
	vector<LineInfo> lines;
	
	void AddLine(const pll &L) {
		while (!lines.empty()) {
			auto &P = lines[SZ(lines) - 1];
			const ll when_takeover = DivFloor(L.nd - P.line.nd, P.line.st - L.st) + 1;
			if (when_takeover <= P.start) {
				lines.pop_back();
			} else {
				lines.push_back({when_takeover, L});
				if (when_takeover <= cur_delta) {
					cur_delta_ptr = SZ(lines) - 1;
				}
				return;
			}
		}
		
		cur_delta_ptr = 0;
		lines.push_back({-kInf, L});
	}
	
	void AddElem(ll value) {
		value += (ll)cur_delta * num_elems;
		pll new_line{-num_elems, value};
		AddLine(new_line);
		++num_elems;
	}
	
	void DecreaseAll(ll amount) {
		cur_delta += amount;
		
		while (cur_delta_ptr + 1 < SZ(lines) && cur_delta >= lines[cur_delta_ptr + 1].start) {
			++cur_delta_ptr;
		}
	}
	
	pll GetOpt() {
		auto &line = lines[cur_delta_ptr];
		return pll{line.Eval(cur_delta), -line.line.st};
	}
	
	friend ostream &operator<<(ostream &os, const DecSeq &ds) {
		os << "{[";
		for (auto line : ds.lines) {
			os << "(" << line.line.st << "," << line.line.nd << ")";
		}
		return os << "] delta=" << ds.cur_delta << "}";
	}
};


string str;
int n, k;


pll SolveWithPenaltyPerIntv(ll penalty_per_intv) {
	const ll mul_coef = n + 1;
	penalty_per_intv = penalty_per_intv * mul_coef + 1;
	//~ debug() << "real penalty" << imie(penalty_per_intv);
	
	vll dp(n + 1);
	ll global_delta = 0;
	vector<DecSeq> seqs(1);
	vll mins_stack;
	dp[0] = penalty_per_intv;
	seqs[0].AddElem(penalty_per_intv);
	mins_stack.push_back(penalty_per_intv);
	
	//~ debug() << "at start";
	//~ debug() << imie(mins_stack) << imie(global_delta);
	//~ debug() << imie(seqs);
	
	for (int i = 0; i < n; ++i) {
		if (str[i] == '(') {
			seqs.emplace_back();
			const ll elem = dp[i] + penalty_per_intv - global_delta;
			seqs.back().AddElem(elem);
			mins_stack.emplace_back(min(mins_stack.back(), elem));
		} else {
			assert(!mins_stack.empty());
			const ll best_top = seqs.back().GetOpt().st;
			seqs.pop_back();
			mins_stack.pop_back();
			
			if (seqs.empty()) {
				seqs.emplace_back();
				seqs[0].AddElem(best_top);
				mins_stack.push_back(best_top);
			} else {
				const int cnt_at_level = seqs.back().num_elems;
				seqs.back().AddElem(best_top);
				global_delta += (ll)cnt_at_level * mul_coef;
				seqs.back().DecreaseAll(mul_coef);
				mins_stack.back() = seqs.back().GetOpt().st;
				if (SZ(mins_stack) >= 2) {
					mini(mins_stack.back(), mins_stack[SZ(mins_stack) - 2]);
				}
			}
		}
		//~ debug() << imie(i) << imie(str[i]) << imie(mins_stack) << imie(global_delta);
		//~ debug() << imie(seqs);
		dp[i + 1] = mins_stack.back() + global_delta;
		//~ debug() << imie(dp[i + 1]);
	}
	
	const ll ans = dp.back();
	return {ans / mul_coef, ans % mul_coef};
}

int main() {
	ios_base::sync_with_stdio(0);
	cin.tie(0);
	cout << fixed << setprecision(11);
	cerr << fixed << setprecision(6);
	
	cin >> n >> k >> str;
	ll min_coef = -1, max_coef = 2e10;
	ll ans = -1e18;
	
	while (max_coef - min_coef > 1) {
		const ll mid_coef = (min_coef + max_coef) / 2;
		auto penalty_sol = SolveWithPenaltyPerIntv(mid_coef);
		
		debug() << imie(mid_coef) << imie(penalty_sol);
		
		if (penalty_sol.nd <= k) {
			ll cand = penalty_sol.st - k * mid_coef;
			ans = max(ans, cand);
			max_coef = mid_coef;
		} else {
			min_coef = mid_coef;
		}
	}
	
	cout << ans << "\n";
}