Problem:
Score : 100 points

Problem Statement

There is a grid with H horizontal rows and W vertical columns.
Let (i, j) denote the square at the i-th row from the top and the j-th column from the left.

For each i and j (1 \leq i \leq H, 1 \leq j \leq W), Square (i, j) is described by a character a_{i, j}.
If a_{i, j} is ., Square (i, j) is an empty square; if a_{i, j} is #, Square (i, j) is a wall square.
It is guaranteed that Squares (1, 1) and (H, W) are empty squares.

Taro will start from Square (1, 1) and reach (H, W) by repeatedly moving right or down to an adjacent empty square.

Find the number of Taro's paths from Square (1, 1) to (H, W).
As the answer can be extremely large, find the count modulo 10^9 + 7.

Constraints

H and W are integers.

2 \leq H, W \leq 1000

a_{i, j} is . or #.

Squares (1, 1) and (H, W) are empty squares.

Input

Input is given from Standard Input in the following format:

H W
a_{1, 1}\ldotsa_{1, W}
:
a_{H, 1}\ldotsa_{H, W}

Output

Print the number of Taro's paths from Square (1, 1) to (H, W), modulo 10^9 + 7.

Sample Input 1

3 4
...#
.#..
....

Sample Output 1

3

There are three paths as follows:

Sample Input 2

5 2
..
#.
..
.#
..

Sample Output 2

0

There may be no paths.

Sample Input 3

5 5
..#..
.....
#...#
.....
..#..

Sample Output 3

24

Sample Input 4

20 20
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................
....................

Sample Output 4

345263555

Be sure to print the count modulo 10^9 + 7.

Solution in Common Lisp (SBCL 1.1.14) (judge verdict: Time Limit Exceeded)

(eval-when (:compile-toplevel :load-toplevel :execute)
  (defparameter OPT
    #+swank '(optimize (speed 3) (safety 2))
    #-swank '(optimize (speed 3) (safety 0) (debug 0)))
  #+swank (progn (ql:quickload '(:cl-debug-print :fiveam))
                 (shadow :run)
                 (use-package :fiveam)))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

;; BEGIN_INSERTED_CONTENTS
;; (with-memoizing (:hash-table :test #'equal :key #'cons)
;;   (defun ...))
;; (with-memoizing (:array (10 10 * 10) :initial-element -1 :element-type 'fixnum)
;;   (defun ...))
(defmacro with-memoizing (cache-attribs def-form)
  (let* ((cache-attribs (if (atom cache-attribs) (list cache-attribs) cache-attribs))
         (cache-type (first cache-attribs))
         (dimensions-with-* (when (eql cache-type :array) (second cache-attribs)))
         (dimensions (remove '* dimensions-with-*))
         (rank (length dimensions))
         (rest-attribs (ecase cache-type
                         (:hash-table (cdr cache-attribs))
                         (:array (cddr cache-attribs))))
         (key (prog1 (getf rest-attribs :key) (remf rest-attribs :key)))
         (cache-form (case cache-type
                       (:hash-table `(make-hash-table ,@rest-attribs))
                       (:array `(make-array ',dimensions ,@rest-attribs))))
         (initial-element (when (eql cache-type :array)
                            (assert (member :initial-element rest-attribs))
                            (getf rest-attribs :initial-element))))
    (let ((value (gensym))
	  (present-p (gensym))
          (name-alias (gensym))
	  (args-lst (gensym))
          (indices (loop repeat rank collect (gensym))))
      (labels ((make-cache-check-form (cache-type args)
                 (case cache-type
                   (:hash-table
                    `(let ((,args-lst (funcall ,(or key #'list) ,@args)))
                       (multiple-value-bind (,value ,present-p)
                           (gethash ,args-lst cache)
                         (if ,present-p
                             ,value
                             (setf (gethash ,args-lst cache)
                                   (,name-alias ,@args))))))
                   (:array
                    (let ((memoized-args (loop for dimension in dimensions-with-*
                                               for arg in args
                                               unless (eql dimension '*)
                                               collect arg)))
                      (if key
                          `(multiple-value-bind ,indices
                               (funcall ,key ,@memoized-args)
                             (let ((,value (aref cache ,@indices)))
                               (if (eql ,initial-element ,value)
                                   (setf (aref cache ,@indices)
                                         (,name-alias ,@args))
                                   ,value)))
                          `(let ((,value (aref cache ,@memoized-args)))
                             (if (eql ,initial-element ,value)
                                 (setf (aref cache ,@memoized-args)
                                       (,name-alias ,@args))
                                 ,value)))))))
               (make-reset-form (cache-type)
                 (case cache-type
                   (:hash-table `(setf cache (make-hash-table ,@rest-attribs)))
                   (:array `(prog1 nil
                              (fill (array-storage-vector cache) ,initial-element)))))
               (make-reset-name (name)
                 (intern (format nil "RESET-~A" (symbol-name name))))
               (extract-declarations (body)
                 (remove-if-not (lambda (form) (eql 'declare (car form))) body)))
        (ecase (car def-form)
          ((defun)
           (destructuring-bind (_ name args &body body) def-form
             (declare (ignore _))
             `(let ((cache ,cache-form))
                (defun ,(make-reset-name name) () ,(make-reset-form cache-type))
                (defun ,name ,args
                  (labels ((,name-alias ,args ,@body))
                    (declare (inline ,name-alias))
                    ,@(extract-declarations body)
                    ,(make-cache-check-form cache-type args))))))
          ((nlet)
           (destructuring-bind (_ name bindings &body body) def-form
             (declare (ignore _))
             `(let ((cache ,cache-form))
                (nlet ,name ,bindings
                      ,(let ((args (mapcar (lambda (x) (if (atom x) x (car x))) bindings)))
                         `(labels ((,name-alias ,args ,@body))
                            (declare (inline ,name-alias))
                            ,@(extract-declarations body)
                            ,(make-cache-check-form cache-type args)))))))
          ((labels flet)
           (destructuring-bind (_ definitions &body labels-body) def-form
             (declare (ignore _))
             (destructuring-bind (name args &body body) (car definitions)
               `(let ((cache ,cache-form))
                  (,(car def-form)
                   ((,(make-reset-name name) () ,(make-reset-form cache-type))
                    (,name ,args
                           (labels ((,name-alias ,args ,@body))
                             (declare (inline ,name-alias))
                             ,@(extract-declarations body)
                             ,(make-cache-check-form cache-type args)))
                    ,@(cdr definitions))
                   (declare (ignorable #',(make-reset-name name)))
                   ,@labels-body))))))))))

;; (test with-memoizing
;;   (finishes (macroexpand `(with-memoizing (:hash-table :test #'equal)
;;                             (defun add (x y) (+ x y)))))
;;   (finishes (macroexpand `(with-memoizing (:array '(10 10)
;;                                            :element-type 'fixnum
;;                                            :initial-element -1)
;;                             (defun add (x y) (+ x y)))))
;;   (finishes (macroexpand `(with-memoizing (:array '(10 10)
;;                                            :element-type 'fixnum
;;                                            :initial-element -1)
;;                             (labels ((add (x y) (+ x y))
;; 		                     (my-print (x) (print x)))
;; 	                      (add 1 2))))))

(defun read-line-into (buf-str &key (in *standard-input*) (terminate-char #\Newline))
  (declare (simple-base-string buf-str))
  (loop for ch of-type base-char =
           #-swank (code-char (read-byte in nil #\Newline))
           #+swank (read-char in nil #\Newline)
        for idx = 0 then (1+ idx)
        until (char= ch #\Newline)
        do (setf (schar buf-str idx) ch)
        finally (when terminate-char (setf (schar buf-str idx) terminate-char))
                (return buf-str)))

(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 4 7 8 15 16 31 32 62 63 64)

(defmacro println (obj &optional (stream '*standard-output*))
  `(let ((*read-default-float-format* 'double-float))
     (prog1 (princ ,obj ,stream) (terpri ,stream))))

;; Hauptteil
(defconstant +magic+ #.(+ 7 (expt 10 9)))
(defun solve (plan)
  (let ((h (array-dimension plan 0))
        (w (array-dimension plan 1)))
    (with-memoizing (:array (1001 1001) :element-type 'uint32
                     :initial-element #.(- (expt 2 32) 1))
      (labels ((recurse (y x)
                 (cond ((= 1 (sbit plan y x)) 0)
                       ((and (zerop y) (zerop x)) 1)
                       ((zerop y)
                        (if (= 0 (sbit plan y x))
                            (recurse y (- x 1))
                            0))
                       ((zerop x)
                        (if (= 0 (sbit plan y x))
                            (recurse (- y 1) x)
                            0))
                       (t
                        (mod
                         (+ (loop for j from (- x 1) downto 0
                                  while (zerop (sbit plan y j))
                                  sum (recurse (- y 1) j))
                            (loop for i from (- y 1) downto 0
                                  while (zerop (sbit plan i x))
                                  sum (recurse i (- x 1))))
                         +magic+)))))
        (recurse (- h 1) (- w 1))))))

(defun main ()
  (let* ((h (read))
         (w (read))
         (plan (make-array (list h w) :element-type 'bit :initial-element 0))
         (buf (make-string w :element-type 'standard-char))
         (horizontal-src-table (make-array h :element-type 'uint32))
         (vertical-src-table (make-array w :element-type 'uint32)))
    (dotimes (y h)
      (let ((line (read-line-into buf :terminate-char nil)))
        (dotimes (x w)
          (when (char= (schar line x) #\#)
            (setf (sbit plan y x) 1)))))
    (dotimes (y h)
      (loop for x from (- w 1) downto 0
            while (zerop (sbit plan y x))
            finally (setf (aref horizontal-src-table y) (+ x 1))))
    (dotimes (x w)
      (loop for y from (- h 1) downto 0
            while (zerop (sbit plan y x))
            finally (setf (aref vertical-src-table x) (+ y 1))))
    (println (solve plan))))

#-swank(main)