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 Julia (1.4.0) (judge verdict: Runtime Error)

const double = Float64
const int = Int64
const vint = Vector{int}
const vdouble = Vector{double}

int(x) = parse(int,x)
parseint(x) = parse(int, x)
readint() = map(parseint, split(readline()))

function main()
  N = readint()[1]

  L=0 # )
  R=0 # (((

  sL=Array{int,2}(undef,N,2) # )))(
  sR=Array{int,2}(undef,N,2)
  nL=0 ; nR =0

  TL = 0
  TR = 0

  # input とまえしょり
  for i in 1:N
    s = readline()
    l = 0
    r = 0
    for j in 1:length(s)
      if s[j]=='('
        r += 1
      else
        if r > 0
          r -= 1
        else
          l +=1
        end
      end
    end
    TL += l
    TR += r

    if l == 0
      R += r
    elseif r == 0
      L += l
    elseif l > r
      if r < L
        L += l-r
      else
        nL += 1
        sL[nL,1] = l
        sL[nL,2] = r
      end
    else
      if l < R
        R += r-l
      else
        nR += 1
        sR[nR,1] =l 
        sR[nR,2] =r
      end

    end
  end
  if TL != TR 
    println("No")
    return 
  end

  sL[1:nL,:] = sL[sortperm(sL[1:nL,2]),:]
  sR[1:nR,:] = sR[sortperm(sR[1:nR,1]),:]

  flag = true
  for i in 1:nL 
    if L >= sL[i,2]
      L += sL[i,1]-sL[i,2]
    else
      frag = false
      break
    end
  end
  for i in 1:nR
    if flag && R >= sR[i,1]
      R += sR[i,2] - sR[i-1]
    else
      flag = false
      break
    end
  end
  if flag 
    println("Yes")
  else
    println("No")
  end

end

main()