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)

data class V(
  val s: String,
  val l: Int,
  val r: Int
)

fun main(args: Array<String>) {
  val N = readInt()
  val V = (1..N).map {
    val s = readString()
    val l = s.count { it == '(' }
    V(s, l, s.length - l)
  }
  val sorted = V.sortedWith(compareBy({ f(it.s) }, { it.r - it.l }, { it.s }))
  val s = buildString {
    sorted.forEach { append(it.s) }
  }
  var num = 0
  for (i in s.indices) {
    if (s[i] == '(') {
      num++
    } else {
      num--
    }
    if (num < 0) {
      println("No")
      return
    }
  }
  println(if (num == 0) "Yes" else "No")
}

fun f(s: String): Int {
  var min = 0
  var num = 0
  for (i in s.indices) {
    if (s[i] == '(') {
      num++
    } else {
      num--
      min = Math.min(min, num)
    }
  }
  return -min
}

fun readString() = readLine()!!
fun readStrings() = readString().split(" ")
fun readInt() = readString().toInt()
fun readLong() = readString().toLong()
fun readInts() = readStrings().map { it.toInt() }
fun readLongs() = readStrings().map { it.toLong() }