Problem:
Score : 1100 points

Problem Statement

We have a grid with N rows and M columns of squares.
Each integer from 1 to NM is written in this grid once.
The number written in the square at the i-th row from the top and the j-th column from the left is A_{ij}.

You need to rearrange these numbers as follows:

First, for each of the N rows, rearrange the numbers written in it as you like.

Second, for each of the M columns, rearrange the numbers written in it as you like.

Finally, for each of the N rows, rearrange the numbers written in it as you like.

After rearranging the numbers, you want the number written in the square at the i-th row from the top and the j-th column from the left to be M\times (i-1)+j.
Construct one such way to rearrange the numbers. The constraints guarantee that it is always possible to achieve the objective.

Constraints

1 \leq N,M \leq 100

1 \leq A_{ij} \leq NM

A_{ij} are distinct.

Input

Input is given from Standard Input in the following format:

N M
A_{11} A_{12} ... A_{1M}
:
A_{N1} A_{N2} ... A_{NM}

Output

Print one way to rearrange the numbers in the following format:

B_{11} B_{12} ... B_{1M}
:
B_{N1} B_{N2} ... B_{NM}
C_{11} C_{12} ... C_{1M}
:
C_{N1} C_{N2} ... C_{NM}

Here B_{ij} is the number written in the square at the i-th row from the top and the j-th column from the left after Step 1, and C_{ij} is the number written in that square after Step 2.

Sample Input 1

3 2
2 6
4 3
1 5

Sample Output 1

2 6
4 3
5 1
2 1
4 3
5 6

Sample Input 2

3 4
1 4 7 10
2 5 8 11
3 6 9 12

Sample Output 2

1 4 7 10
5 8 11 2
9 12 3 6
1 4 3 2
5 8 7 6
9 12 11 10

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

(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-int:defconstant-eqx opt
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0))
    #'equal)
  #+swank (ql:quickload '(:cl-debug-print :fiveam) :silent t)
  #-swank (set-dispatch-macro-character
           #\# #\> (lambda (s c p) (declare (ignore c p)) `(values ,(read s nil nil t)))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

(defmacro define-int-types (&rest bits)
  `(progn
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "UINT~A" b)) () '(unsigned-byte ,b))) bits)
     ,@(mapcar (lambda (b) `(deftype ,(intern (format nil "INT~A" b)) () '(signed-byte ,b))) bits)))
(define-int-types 2 4 7 8 15 16 31 32 62 63 64)

;; BEGIN_INSERTED_CONTENTS
(declaim (ftype (function * (values fixnum &optional)) read-fixnum))
(defun read-fixnum (&optional (in *standard-input*))
  "NOTE: cannot read -2^62"
  (declare #.opt)
  (macrolet ((%read-byte ()
               `(the (unsigned-byte 8)
                     #+swank (char-code (read-char in nil #\Nul))
                     #-swank (sb-impl::ansi-stream-read-byte in nil #.(char-code #\Nul) nil))))
    (let* ((minus nil)
           (result (loop (let ((byte (%read-byte)))
                           (cond ((<= 48 byte 57)
                                  (return (- byte 48)))
                                 ((zerop byte) ; #\Nul
                                  (error "Read EOF or #\Nul."))
                                 ((= byte #.(char-code #\-))
                                  (setq minus t)))))))
      (declare ((integer 0 #.most-positive-fixnum) result))
      (loop
        (let* ((byte (%read-byte)))
          (if (<= 48 byte 57)
              (setq result (+ (- byte 48)
                              (* 10 (the (integer 0 #.(floor most-positive-fixnum 10))
                                         result))))
              (return (if minus (- result) result))))))))

(in-package :cl-user)

(defmacro dbg (&rest forms)
  #+swank
  (if (= (length forms) 1)
      `(format *error-output* "~A => ~A~%" ',(car forms) ,(car forms))
      `(format *error-output* "~A => ~A~%" ',forms `(,,@forms)))
  #-swank (declare (ignore forms)))

(declaim (inline println))
(defun println (obj &optional (stream *standard-output*))
  (let ((*read-default-float-format* 'double-float))
    (prog1 (princ obj stream) (terpri stream))))

(defconstant +mod+ 1000000007)

;;;
;;; Body
;;;

(defun solve (n m as)
  (declare #.opt
           ((simple-array uint31 (100 100)) as)
           (uint8 n m))
  (let ((counts (make-array '(100 100) :element-type 'uint31 :initial-element 0)))
    (dotimes (i n)
      (dotimes (j m)
        (let ((a (aref as i j)))
          (incf (aref counts j (floor a m))))))
    (let ((score 0))
      (declare (uint31 score))
      (dotimes (i m)
        (dotimes (j n)
          (incf score (abs (- (aref counts i j) 1)))))
      (loop (when (zerop score)
              (return as))
            (let* ((y (random n))
                   (x1 (random m))
                   (x2 (random m))
                   (a1 (floor (aref as y x1) m))
                   (a2 (floor (aref as y x2) m))
                   (delta (+ (if (= 0 (aref counts x1 a2)) -1 1)
                             (if (= 1 (aref counts x1 a1)) 1 -1)
                             (if (= 0 (aref counts x2 a1)) -1 1)
                             (if (= 1 (aref counts x2 a2)) 1 -1))))
              (when (<= delta 0)
                (incf score delta)
                (incf (aref counts x1 a2))
                (incf (aref counts x2 a1))
                (decf (aref counts x1 a1))
                (decf (aref counts x2 a2))
                (rotatef (aref as y x1) (aref as y x2))))))))

(defconstant +nan+ #x7fffffff)
(defun main ()
  (declare #.opt)
  (let* ((n (read))
         (m (read))
         (as (make-array '(100 100) :element-type 'uint31 :initial-element 0)))
    (declare (uint8 n m)
             ((simple-array uint31 (* *)) as))
    (dotimes (i n)
      (dotimes (j m)
        (setf (aref as i j) (- (read-fixnum) 1))))
    (solve n m as)
    (write-string
     (with-output-to-string (*standard-output* nil :element-type 'base-char)
       (dotimes (i n)
         (let ((init t))
           (dotimes (j m (terpri))
             (if init
                 (setq init nil)
                 (write-char #\ ))
             (princ (aref as i j)))))
       (dotimes (j m)
         (dotimes (i1 n)
           (loop for i2 from (+ i1 1) below n
                 when (> (aref as i1 j) (aref as i2 j))
                 do (rotatef (aref as i1 j) (aref as i2 j)))))
       (dotimes (i n)
         (let ((init t))
           (dotimes (j m (terpri))
             (if init
                 (setq init nil)
                 (write-char #\ ))
             (princ (aref as i j)))))))))

(defun test (n m sample)
  (dotimes (_ sample)
    (let ((as (make-array (list n m) :element-type 'uint31 :initial-element 0)))
      (dotimes (i (* n m))
        (setf (aref (array-storage-vector as) i) i))
      (shuffle! (array-storage-vector as))
      (solve as))))
#-swank (main)

;;;
;;; Test and benchmark
;;;

#+swank
(defun get-clipbrd ()
  (with-output-to-string (out)
    #+os-windows (run-program "powershell.exe" '("-Command" "Get-Clipboard") :output out :search t)
    #+os-unix (run-program "xsel" '("-b" "-o") :output out :search t)))

#+swank (defparameter *this-pathname* (uiop:current-lisp-file-pathname))
#+swank (defparameter *dat-pathname* (uiop:merge-pathnames* "test.dat" *this-pathname*))

#+swank
(defun run (&optional thing (out *standard-output*))
  "THING := null | string | symbol | pathname

null: run #'MAIN using the text on clipboard as input.
string: run #'MAIN using the string as input.
symbol: alias of FIVEAM:RUN!.
pathname: run #'MAIN using the text file as input."
  (let* ((*standard-output* (or out (make-string-output-stream)))
         (res (etypecase thing
                (null
                 (with-input-from-string (*standard-input* (delete #\Return (get-clipbrd)))
                   (main)))
                (string
                 (with-input-from-string (*standard-input* (delete #\Return thing))
                   (main)))
                (symbol (5am:run! thing))
                (pathname
                 (with-open-file (*standard-input* thing)
                   (main))))))
    (if out res (get-output-stream-string *standard-output*))))

#+swank
(defun gen-dat ()
  (uiop:with-output-file (out *dat-pathname* :if-exists :supersede)
    (format out "")))

#+swank
(defun bench (&optional (out (make-broadcast-stream)))
  (time (run *dat-pathname* out)))

;; To run: (5am:run! :sample)
#+swank
(it.bese.fiveam:test :sample
  (it.bese.fiveam:is
   (equal "2 6 
4 3 
5 1 
2 1 
4 3 
5 6
"
    (run "3 2
2 6
4 3
1 5
" nil)))
  (it.bese.fiveam:is
   (equal "1 4 7 10 
5 8 11 2 
9 12 3 6 
1 4 3 2 
5 8 7 6 
9 12 11 10
"
    (run "3 4
1 4 7 10
2 5 8 11
3 6 9 12
" nil))))