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 Common Lisp (SBCL 2.0.3) (judge verdict: Wrong Answer)

#-swank
(unless (member :child-sbcl *features*)
  "I refered from https://competitive12.blogspot.com/2020/03/common-lisp.html thank you!"
  (quit
   :unix-status
   (process-exit-code
    (run-program *runtime-pathname*
                 `("--control-stack-size" "128MB"
                   "--noinform" "--disable-ldb" "--lose-on-corruption" "--end-runtime-options"
                   "--eval" "(push :child-sbcl *features*)"
                   "--script" ,(namestring *load-pathname*))
                 :output t :error t :input t))))
(defmacro dp (name args memo-size &body body)
  (let ((memo (gensym)))
    `(let ((,memo (make-array ,memo-size :initial-element nil)))
       (defun ,name ,args
         (or (aref ,memo ,@args)
             (setf (aref ,memo ,@args)
                   ,@body))))))

(defun split (x str)
  (let ((pos (search x str))
        (size (length x)))
    (if pos
        (cons (subseq str 0 pos)
              (split x (subseq str (+ pos size))))
        (list str))))

(defmacro collect-times (time body)
  `(loop repeat ,time collect ,body))

(defun read-times (time)
  (collect-times time (read)))

(defmacro aif (expr then else)
  `(let ((it ,expr))
     (if it ,then ,else)))

(defmacro awhen (expr &rest then)
  `(aif ,expr (progn ,@then) nil))

(defun comulative (function list &key base)
  (if base
      (comulative function (cons base list))
      (do ((lst (cdr list) (cdr lst))
           (acc (list (car list))))
          ((null lst) (reverse acc))
        (push (funcall function (car acc) (car lst)) acc))))

(defun binary-search (function left right
                      &optional
                        (eps 1)
                        (average-function (lambda (x y) (ash (+ x y) -1))))
  (if (<= (abs (- left right)) eps)
      right
      (let ((mid (funcall average-function left right)))
        (if (funcall function mid)
            (binary-search function left mid eps average-function)
            (binary-search function mid right eps average-function)))))

(defun arithmetic-mean (&rest body)
  (/ (apply #'+ body) (length body)))

(defvar +MOD+ (+ (expt 10 9) 7))

(defun mapa-b (fn a b &optional (step 1))
  (do ((i a (+ i step))
       (result nil))
      ((> i b) (nreverse result))
    (push (funcall fn i) result)))

(defun map0-n (fn n)
  (mapa-b fn 0 n))

(defun map1-n (fn n)
  (mapa-b fn 1 n))

(defun read-string (&optional (stream *standard-input*))
  (labels ((is-empty (x)
             (or (char= x #\space) (char= x #\newline))))
    (loop for char = (read-char stream)
          with result
          when (is-empty char)
            do (return (concatenate 'string (nreverse result)))
          do (push char result))))

(defun merge-sort (lst &optional (compare #'<))
  (let ((turn 0))
    (labels ((merge-list (a b a-length b-length)
               (cond ((zerop a-length) b)
                     ((zerop b-length) a)
                     ((funcall compare (car b) (car a))
                      (incf turn a-length)
                      (cons (car b)
                            (merge-list a (cdr b) a-length (1- b-length))))
                     (t
                      (cons (car a)
                            (merge-list (cdr a) b (1- a-length) b-length)))))
             (f (lst length)
               (if (= length 1)
                   lst
                   (let ((mid (ash length -1)))
                     (merge-list (f (subseq lst 0 mid) mid)
                                 (f (subseq lst mid) (- length mid))
                                 mid
                                 (- length mid))))))
      (values (f lst (length lst)) turn))))

(defun group (lst &optional (test #'eql) (key nil))
  (let ((table (make-hash-table :test test)))
    (mapc (lambda (x)
            (push x (gethash (if key (funcall key x) x) table)))
          lst)
    (loop for value being each hash-value in table
          collect value)))

(defun parenthese (str)
  (let ((sum 0)
        (min 0))
    (loop for i across str
          do
             (progn
               (if (char= i #\()
                   (incf sum)
                   (progn
                     (decf sum)
                     (setf min (min min sum))))))
    (cons min sum)))


(defun main (strs)
  (let* ((nums (mapcar #'parenthese strs))
         (adder (sort (remove-if (lambda (x) (< (cdr x) 0)) nums)
                      (lambda (x y)
                        (or (> (car x) (car y)) (and (= (car x) (car y)) (> (cdr x) (cdr y)))))))
         (subber (sort (remove-if (lambda (x) (>= (cdr x) 0)) nums)
                       (lambda (x y)
                         (or (< (car x) (car y)) (and (= (car x) (car y)) (> (cdr x) (cdr y)))))))
         (sum 0))
    (loop for i in adder
          for val = (car i)
          for add = (cdr i)
          when (< (+ sum val) 0)
            do (return-from main nil)
          do
             (incf sum add))
    (loop for i in subber
          for val = (car i)
          for add = (cdr i)
          when (< (+ sum val) 0)
            do (return-from main nil)
          do
             (incf sum add))
    (zerop sum)))


(princ (if (main (collect-times (read) (read-string))) "Yes" "No"))