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 OCaml (4.10.0) (judge verdict: Wrong Answer)


let parse s =
  Array.init (String.length s) (fun i -> s.[i])
  |> Array.fold_left (fun (p, b) c ->
      if c = '(' then (p+1, b) else (p-1, min b (p-1))) (0, 0)

let () =
  Scanf.scanf "%d" @@ fun n ->
  let s = Array.init n (fun _ -> Scanf.scanf " %s" parse) in
  Array.sort (fun (a, b) (c, d) -> d - b) s;
  let tp, tb =
    Array.fold_left (fun (tp, tb) (p, b) ->
      (tp + p, min tb (tp + b))) (0, 0) s in

  Printf.printf "%d %d\n" tp tb;
  let ans = if tp = 0 && tb >= 0 then "Yes" else "No" in
  print_endline ans