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 Kotlin (1.3.71) (judge verdict: Wrong Answer)

import java.util.*

fun main() {
    fun log(str: String) {
//        println(str)
    }
    val sc = Scanner(System.`in`)

    data class LR(val l: Int, val r: Int)

    fun run (): Boolean {
        val n = sc.nextInt()

        val a = ArrayList<LR>(n)

        for (i in 0 until n) {
            val s = sc.next()
            var l = 0 // (
            var r = 0 // )
            for (c in s) {
                if (c == '(') {
                    l++
                } else {
                    if (l > 0) {
                        l--
                    } else {
                        r++
                    }
                }
            }
            if (l != 0 || r != 0)
                a.add(LR(l, r))
        }

        if (a.sumBy { it.l } != a.sumBy { it.r }) {
            log("sumby")
            return false
        } else {
            var c = 0
            log(a.joinToString("\n"))
            val a1 = a.toMutableList().apply { sortWith(compareBy<LR> { -it.l }.thenBy { it.r }) }
            while (a1.isNotEmpty()) {
                val size = a1.size
                val i = a1.iterator()

                while (i.hasNext()) {
                    val lr = i.next()
                    if (c - lr.r < 0) {
                        // can't add
                        log("can't add: $c, $lr")
                        continue
                    }
                    log("add: $c, $lr")
                    c -= lr.r
                    c += lr.l
                    i.remove()
                }

                if (size == a1.size) {
                    return false
                }
            }
            if (c == 0) {
                return true
            } else {
                log(a.joinToString("\n"))
                return false
            }
        }
    }
    if (run()) {
        println("Yes")
    } else {
        println("No")
    }
}