Problem:
Score : 600 points

Problem Statement

A bracket sequence is a string that is one of the following:

An empty string;

The concatenation of (, A, and ) in this order, for some bracket sequence A ;

The concatenation of A and B in this order, for some non-empty bracket sequences A and B /

Given are N strings S_i. Can a bracket sequence be formed by concatenating all the N strings in some order?

Constraints

1 \leq N \leq 10^6

The total length of the strings S_i is at most 10^6.

S_i is a non-empty string consisting of ( and ).

Input

Input is given from Standard Input in the following format:

N
S_1
:
S_N

Output

If a bracket sequence can be formed by concatenating all the N strings in some order, print Yes; otherwise, print No.

Sample Input 1

2
)
(()

Sample Output 1

Yes

Concatenating (() and ) in this order forms a bracket sequence.

Sample Input 2

2
)(
()

Sample Output 2

No

Sample Input 3

4
((()))
((((((
))))))
()()()

Sample Output 3

Yes

Sample Input 4

3
(((
)
)

Sample Output 4

No

Solution in C# (Mono-mcs 6.8.0.105) (judge verdict: Runtime Error)

using System;
using System.Collections.Generic;
using System.IO;
using System.Linq;
using static System.Math;
using System.Text;
using System.Threading;
using System.Globalization;
using System.Runtime.CompilerServices;
using Library;

namespace Program
{
    public static class ABC167F
    {
        static bool SAIKI = false;
        static public int numberOfRandomCases = 0;
        static public void MakeTestCase(List<string> _input, List<string> _output, ref Func<string[], bool> _outputChecker)
        {
        }
        public struct Info
        {
            public long zoubun;
            public long depth;
            public long idx;
        }
        static public void Solve()
        {
            var N = NN;
            var SList = NSList(N);
            var info = new Info[N];
            for (var i = 0; i < N; i++)
            {
                var x = 0L;
                var minx = 0L;
                foreach (var item in SList[i])
                {
                    if (item == '(') ++x;
                    else --x;
                    minx = Min(minx, x);
                }
                info[i] = new Info { zoubun = x, depth = minx, idx = i };
            }
            var sum = info.Sum(e => e.zoubun);
            if (sum == 0)
            {
                var list = new LIB_Deque<Info>();
                var left = 0L;
                var min = 0L;
                foreach (var item in info.OrderBy(e => e.depth).ThenBy(e => Abs(e.zoubun)))
                {
                    if (item.zoubun > 0)
                    {
                        list.PushFront(item);
                        left -= item.zoubun;
                    }
                    else if (item.zoubun < 0)
                    {
                        list.PushBack(item);
                    }
                    else
                    {
                        min = Min(min, item.depth);
                    }
                }
                if (left > min)
                {
                    Console.WriteLine("No");
                    return;
                }
                var now = 0L;
                for (var i = 0; i < list.Count; i++)
                {
                    var infoo = list[i];
                    var s = SList[infoo.idx];
                    foreach (var item in s)
                    {
                        if (item == '(') ++now;
                        else --now;
                        if (now < 0)
                        {
                            throw new Exception();
                            Console.WriteLine("No");
                            return;
                        }
                    }
                }
                Console.WriteLine("Yes");
            }
            else
            {
                Console.WriteLine("No");
            }
        }
        class Printer : StreamWriter
        {
            public override IFormatProvider FormatProvider { get { return CultureInfo.InvariantCulture; } }
            public Printer(Stream stream) : base(stream, new UTF8Encoding(false, true)) { base.AutoFlush = false; }
            public Printer(Stream stream, Encoding encoding) : base(stream, encoding) { base.AutoFlush = false; }
        }
        static LIB_FastIO fastio = new LIB_FastIODebug();
        static public void Main(string[] args) { if (args.Length == 0) { fastio = new LIB_FastIO(); Console.SetOut(new Printer(Console.OpenStandardOutput())); } if (SAIKI) { var t = new Thread(Solve, 134217728); t.Start(); t.Join(); } else Solve(); Console.Out.Flush(); }
        static long NN => fastio.Long();
        static double ND => fastio.Double();
        static string NS => fastio.Scan();
        static long[] NNList(long N) => Repeat(0, N).Select(_ => NN).ToArray();
        static double[] NDList(long N) => Repeat(0, N).Select(_ => ND).ToArray();
        static string[] NSList(long N) => Repeat(0, N).Select(_ => NS).ToArray();
        static long Count<T>(this IEnumerable<T> x, Func<T, bool> pred) => Enumerable.Count(x, pred);
        static IEnumerable<T> Repeat<T>(T v, long n) => Enumerable.Repeat<T>(v, (int)n);
        static IEnumerable<int> Range(long s, long c) => Enumerable.Range((int)s, (int)c);
        static IOrderedEnumerable<T> OrderByRand<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x, _ => xorshift);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x) => Enumerable.OrderBy(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderBy<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderBy(x.OrderByRand(), selector);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x) => Enumerable.OrderByDescending(x.OrderByRand(), e => e);
        static IOrderedEnumerable<T1> OrderByDescending<T1, T2>(this IEnumerable<T1> x, Func<T1, T2> selector) => Enumerable.OrderByDescending(x.OrderByRand(), selector);
        static IOrderedEnumerable<string> OrderBy(this IEnumerable<string> x) => x.OrderByRand().OrderBy(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderBy<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderBy(selector, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<string> OrderByDescending(this IEnumerable<string> x) => x.OrderByRand().OrderByDescending(e => e, StringComparer.OrdinalIgnoreCase);
        static IOrderedEnumerable<T> OrderByDescending<T>(this IEnumerable<T> x, Func<T, string> selector) => x.OrderByRand().OrderByDescending(selector, StringComparer.OrdinalIgnoreCase);
        static uint xorshift { get { _xsi.MoveNext(); return _xsi.Current; } }
        static IEnumerator<uint> _xsi = _xsc();
        static IEnumerator<uint> _xsc() { uint x = 123456789, y = 362436069, z = 521288629, w = (uint)(DateTime.Now.Ticks & 0xffffffff); while (true) { var t = x ^ (x << 11); x = y; y = z; z = w; w = (w ^ (w >> 19)) ^ (t ^ (t >> 8)); yield return w; } }
    }
}
namespace Library {
    class LIB_Deque<T>
    {
        T[] array;
        int front, cap;
        public int Count;
        public T this[long i]
        {
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            get { return array[GetIndex((int)i)]; }
            [MethodImpl(MethodImplOptions.AggressiveInlining)]
            set { array[GetIndex((int)i)] = value; }
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public LIB_Deque(long cap = 16)
        {
            array = new T[this.cap = (int)cap];
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        int GetIndex(int i)
        {
            if (i >= cap) throw new Exception();
            var r = front + i;
            return r >= cap ? r - cap : r;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void PushFront(T x)
        {
            if (Count == cap) Extend();
            if (--front < 0) front += array.Length;
            array[front] = x;
            ++Count;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T PopFront()
        {
            if (Count-- == 0) throw new Exception();
            var r = array[front++];
            if (front >= cap) front -= cap;
            return r;
        }
        public T Front => array[front];
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void PushBack(T x)
        {
            if (Count == cap) Extend();
            var i = front + Count++;
            array[i >= cap ? i - cap : i] = x;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T PopBack()
        {
            if (Count == 0) throw new Exception();
            return array[GetIndex(--Count)];
        }
        public T Back => array[GetIndex(Count - 1)];
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        void Extend()
        {
            T[] nb = new T[cap << 1];
            if (front > cap - Count)
            {
                var l = array.Length - front; Array.Copy(array, front, nb, 0, l);
                Array.Copy(array, 0, nb, l, Count - l);
            }
            else Array.Copy(array, front, nb, 0, Count);
            array = nb; front = 0; cap <<= 1;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public void Insert(long i, T x)
        {
            if (i > Count) throw new Exception();
            this.PushFront(x);
            for (int j = 0; j < i; j++) this[j] = this[j + 1];
            this[i] = x;
        }
        [MethodImpl(MethodImplOptions.AggressiveInlining)]
        public T RemoveAt(long i)
        {
            if (i < 0 || i >= Count) throw new Exception();
            var r = this[i];
            for (var j = i; j > 0; j--) this[j] = this[j - 1];
            this.PopFront();
            return r;
        }
    }
    class LIB_FastIO
    {
        public LIB_FastIO() { str = Console.OpenStandardInput(); }
        readonly Stream str;
        readonly byte[] buf = new byte[1024];
        int len, ptr;
        public bool isEof = false;
        public bool IsEndOfStream { get { return isEof; } }
        byte read()
        {
            if (isEof) throw new EndOfStreamException();
            if (ptr >= len)
            {
                ptr = 0;
                if ((len = str.Read(buf, 0, 1024)) <= 0)
                {
                    isEof = true;
                    return 0;
                }
            }
            return buf[ptr++];
        }
        char Char()
        {
            byte b = 0;
            do b = read();
            while (b < 33 || 126 < b);
            return (char)b;
        }
        virtual public string Scan()
        {
            var sb = new StringBuilder();
            for (var b = Char(); b >= 33 && b <= 126; b = (char)read())
                sb.Append(b);
            return sb.ToString();
        }
        virtual public long Long()
        {
            long ret = 0; byte b = 0; var ng = false;
            do b = read();
            while (b != '-' && (b < '0' || '9' < b));
            if (b == '-') { ng = true; b = read(); }
            for (; true; b = read())
            {
                if (b < '0' || '9' < b)
                    return ng ? -ret : ret;
                else ret = ret * 10 + b - '0';
            }
        }
        virtual public double Double() { return double.Parse(Scan(), CultureInfo.InvariantCulture); }
    }
    class LIB_FastIODebug : LIB_FastIO
    {
        Queue<string> param = new Queue<string>();
        string NextString() { if (param.Count == 0) foreach (var item in Console.ReadLine().Split(' ')) param.Enqueue(item); return param.Dequeue(); }
        public LIB_FastIODebug() { }
        public override string Scan() => NextString();
        public override long Long() => long.Parse(NextString());
        public override double Double() => double.Parse(NextString());
    }
}