Problem: This is a harder version of the problem. In this version, n ≤ 300 000.

Vasya is an experienced developer of programming competitions' problems. As all great minds at some time, Vasya faced a creative crisis. To improve the situation, Petya gifted him a string consisting of opening and closing brackets only. Petya believes, that the beauty of the bracket string is a number of its cyclical shifts, which form a correct bracket sequence.

To digress from his problems, Vasya decided to select two positions of the string (not necessarily distinct) and swap characters located at this positions with each other. Vasya will apply this operation exactly once. He is curious what is the maximum possible beauty he can achieve this way. Please help him.

We remind that bracket sequence s is called correct if: 

  * s is empty; 
  * s is equal to "(t)", where t is correct bracket sequence; 
  * s is equal to t_1 t_2, i.e. concatenation of t_1 and t_2, where t_1 and t_2 are correct bracket sequences. 



For example, "(()())", "()" are correct, while ")(" and "())" are not.

The cyclical shift of the string s of length n by k (0 ≤ k < n) is a string formed by a concatenation of the last k symbols of the string s with the first n - k symbols of string s. For example, the cyclical shift of string "(())()" by 2 equals "()(())".

Cyclical shifts i and j are considered different, if i ≠ j.

Input

The first line contains an integer n (1 ≤ n ≤ 300 000), the length of the string.

The second line contains a string, consisting of exactly n characters, where each of the characters is either "(" or ")".

Output

The first line should contain a single integer — the largest beauty of the string, which can be achieved by swapping some two characters.

The second line should contain integers l and r (1 ≤ l, r ≤ n) — the indices of two characters, which should be swapped in order to maximize the string's beauty.

In case there are several possible swaps, print any of them.

Examples

Input


10
()()())(()


Output


5
8 7


Input


12
)(()(()())()


Output


4
5 10


Input


6
)))(()


Output


0
1 1

Note

In the first example, we can swap 7-th and 8-th character, obtaining a string "()()()()()". The cyclical shifts by 0, 2, 4, 6, 8 of this string form a correct bracket sequence.

In the second example, after swapping 5-th and 10-th character, we obtain a string ")(())()()(()". The cyclical shifts by 11, 7, 5, 3 of this string form a correct bracket sequence.

In the third example, swap of any two brackets results in 0 cyclical shifts being correct bracket sequences. 

Solution (in JAVA):

Solution:
import java.io.OutputStream;
import java.io.IOException;
import java.io.InputStream;
import java.io.OutputStream;
import java.io.BufferedWriter;
import java.io.IOException;
import java.io.InputStreamReader;
import java.util.Stack;
import java.io.UncheckedIOException;
import java.nio.charset.Charset;
import java.util.StringTokenizer;
import java.io.Writer;
import java.io.OutputStreamWriter;
import java.io.BufferedReader;
import java.io.InputStream;

/**
 * Built using CHelper plug-in
 * Actual solution is at the top
 *
 * @author mikit
 */
public class Main {
    public static void main(String[] args) {
        InputStream inputStream = System.in;
        OutputStream outputStream = System.out;
        LightScanner in = new LightScanner(inputStream);
        LightWriter out = new LightWriter(outputStream);
        BTheWorldIsJustAProgrammingTaskHardVersion solver = new BTheWorldIsJustAProgrammingTaskHardVersion();
        solver.solve(1, in, out);
        out.close();
    }

    static class BTheWorldIsJustAProgrammingTaskHardVersion {
        public void solve(int testNumber, LightScanner in, LightWriter out) {
            // out.setBoolLabel(LightWriter.BoolLabel.YES_NO_FIRST_UP);
            int n = in.ints();
            char[] s = in.string().toCharArray();
            int close = 0;
            for (int i = 0; i < n; i++) close += s[i] - '(';
            if (close * 2 != n) {
                out.ans(0).ln()
                        .ans(1).ans(1).ln();
                return;
            }


            int base = 0;
            int[] depth = new int[n], another = new int[n];
            int[][] sum = new int[2][2 * n + 1];
            {
                int d = 0;
                for (int i = 0; i < n; i++) {
                    if (s[i] == '(') d++;
                    else d--;
                    base = Math.min(base, d);
                    depth[i] = d;
                }
            }
            {
                int offset = -1;
                for (int i = 0; i < n; i++) {
                    if (depth[i] - base == 0) {
                        offset = (i + 1) % n;
                        break;
                    }
                }
                Stack<Integer> st = new Stack<>();
                for (int j = 0; j < n; j++) {
                    int i = (offset + j) % n;
                    if (s[i] == '(') st.push(i);
                    else {
                        int a = st.pop();
                        another[a] = i < a ? i + n : i;
                        another[i] = a;
                    }
                }
            }


            {
                for (int i = 0; i < 2 * n; i++) {
                    if (s[i % n] == '(' || depth[i % n] - base != 1) {
                        sum[0][i + 1] = sum[0][i];
                    } else sum[0][i + 1] = sum[0][i] + 1;
                    if (s[i % n] == '(' || depth[i % n] - base != 2) {
                        sum[1][i + 1] = sum[1][i];
                    } else sum[1][i + 1] = sum[1][i] + 1;
                }
            }

            int a = 0, b = 0, ans = eval(s, a, b);
            {
                int cnt = -1, x = 0, y = 0;
                for (int i = 0; i < n; i++) {
                    if (s[i] == '(' && depth[i] - base == 1) {
                        int c = sum[0][another[i]] - sum[0][i + 1];
                        if (c > cnt) {
                            cnt = c;
                            x = i;
                            y = another[i] % n;
                        }
                    }
                }
                if (cnt >= 0) {
                    int t = eval(s, x, y);
                    if (t > ans) {
                        ans = t;
                        a = x;
                        b = y;
                    }
                }
            }
            {
                //System.out.println(Arrays.toString(sum[1]));
                int cnt = -1, x = 0, y = 0;
                for (int i = 0; i < n; i++) {
                    if (s[i] == '(' && depth[i] - base == 2) {
                        int c = sum[1][another[i]] - sum[1][i + 1];
                        //System.out.println("FOUND: " + i + "~" + (another[i]) + ":" + (sum[1][another[i]] - sum[1][i + 1]));
                        if (c > cnt) {
                            cnt = c;
                            x = i;
                            y = another[i] % n;
                        }
                    }
                }
                if (cnt >= 0) {
                    int t = eval(s, x, y);
                    if (t > ans) {
                        ans = t;
                        a = x;
                        b = y;
                    }
                }
            }


            out.ans(ans).ln()
                    .ans(Math.min(a, b) + 1).ans(Math.max(a, b) + 1).ln();

            //System.out.println(Arrays.toString(cnt));
        }

        private static void swap(char[] s, int x, int y) {
            char t = s[x];
            s[x] = s[y];
            s[y] = t;
        }

        private static int eval(char[] s, int x, int y) {
            swap(s, x, y);
            int base = 0;
            {
                int d = 0;
                for (char c : s) {
                    if (c == '(') d++;
                    else d--;
                    base = Math.min(base, d);
                }
            }

            int ans = 0;
            {
                int d = 0;
                for (char c : s) {
                    if (c == '(') d++;
                    else d--;
                    if (c == ')' && d - base == 0) ans++;
                }
            }
            swap(s, x, y);
            return ans;
        }

    }

    static class LightWriter implements AutoCloseable {
        private final Writer out;
        private boolean autoflush = false;
        private boolean breaked = true;

        public LightWriter(Writer out) {
            this.out = out;
        }

        public LightWriter(OutputStream out) {
            this(new BufferedWriter(new OutputStreamWriter(out, Charset.defaultCharset())));
        }

        public LightWriter print(char c) {
            try {
                out.write(c);
                breaked = false;
            } catch (IOException ex) {
                throw new UncheckedIOException(ex);
            }
            return this;
        }

        public LightWriter print(String s) {
            try {
                out.write(s, 0, s.length());
                breaked = false;
            } catch (IOException ex) {
                throw new UncheckedIOException(ex);
            }
            return this;
        }

        public LightWriter ans(String s) {
            if (!breaked) {
                print(' ');
            }
            return print(s);
        }

        public LightWriter ans(int i) {
            return ans(Integer.toString(i));
        }

        public LightWriter ln() {
            print(System.lineSeparator());
            breaked = true;
            if (autoflush) {
                try {
                    out.flush();
                } catch (IOException ex) {
                    throw new UncheckedIOException(ex);
                }
            }
            return this;
        }

        public void close() {
            try {
                out.close();
            } catch (IOException ex) {
                throw new UncheckedIOException(ex);
            }
        }

    }

    static class LightScanner {
        private BufferedReader reader = null;
        private StringTokenizer tokenizer = null;

        public LightScanner(InputStream in) {
            reader = new BufferedReader(new InputStreamReader(in));
        }

        public String string() {
            if (tokenizer == null || !tokenizer.hasMoreTokens()) {
                try {
                    tokenizer = new StringTokenizer(reader.readLine());
                } catch (IOException e) {
                    throw new UncheckedIOException(e);
                }
            }
            return tokenizer.nextToken();
        }

        public int ints() {
            return Integer.parseInt(string());
        }

    }
}