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 kotlin.math.min

fun main() = abc167f()

@OptIn(ExperimentalStdlibApi::class)
fun abc167f() {
    data class Bracket(val lowestHeight: Int, val transition: Int)

    val n = readLine()!!.toInt()
    val brackets = (1..n).map {
        val s = readLine()!!
        var currHeight = 0
        var lowestHeight = 0
        for (c in s) {
            when (c) {
                '(' -> currHeight++
                else -> currHeight--
            }
            lowestHeight = min(lowestHeight, currHeight)
        }
        Bracket(lowestHeight, currHeight)
    }

    var currHeight = 0
    val (positives, negatives) = brackets.partition { it.transition >= 0 }

    fun simulate(list: List<Bracket>): Boolean {
        for ((lowestHeight, transition) in list) {
            if (currHeight + lowestHeight < 0) return false
            currHeight += transition
        }
        return true
    }

    if (!simulate(positives.sortedByDescending { it.lowestHeight })) return println("No")
    if (!simulate(negatives.sortedByDescending { it.transition })) return println("No")

    val answer = if (currHeight == 0) "Yes" else "No"

    println(answer)
}