Problem:
Score : 600 points

Problem Statement

There is an arithmetic progression with L terms: s_0, s_1, s_2, ... , s_{L-1}.

The initial term is A, and the common difference is B. That is, s_i = A + B \times i holds.

Consider the integer obtained by concatenating the terms written in base ten without leading zeros. For example, the sequence 3, 7, 11, 15, 19 would be concatenated into 37111519. What is the remainder when that integer is divided by M?

Constraints

All values in input are integers.

1 \leq L, A, B < 10^{18}

2 \leq M \leq 10^9

All terms in the arithmetic progression are less than 10^{18}.

Input

Input is given from Standard Input in the following format:

L A B M

Output

Print the remainder when the integer obtained by concatenating the terms is divided by M.

Sample Input 1

5 3 4 10007

Sample Output 1

5563

Our arithmetic progression is 3, 7, 11, 15, 19, so the answer is 37111519 mod 10007, that is, 5563.

Sample Input 2

4 8 1 1000000

Sample Output 2

891011

Sample Input 3

107 10000000000007 1000000000000007 998244353

Sample Output 3

39122908

Solution in Common Lisp (SBCL 1.1.14) (judge verdict: Accepted)

;; -*- coding: utf-8 -*-
(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)))
  #-swank (set-dispatch-macro-character #\# #\> (lambda (s c p) (declare (ignore c p)) (read s nil (values) t))))
#+swank (cl-syntax:use-syntax cl-debug-print:debug-print-syntax)

;; BEGIN_INSERTED_CONTENTS
;;;
;;; ARRAY-ELEMENT-TYPE is not constant-folded on SBCL version earlier than
;;; 1.5.0. See
;;; https://github.com/sbcl/sbcl/commit/9f0d12e7ab961828931d01c0b2a76a5885ad35d2
;;;

(eval-when (:compile-toplevel :load-toplevel :execute)
  (sb-c:deftransform array-element-type ((array))
    (let ((type (sb-c::lvar-type array)))
      (flet ((element-type (type)
               (and (sb-c::array-type-p type)
                    (sb-int:neq (sb-kernel::array-type-specialized-element-type type) sb-kernel:*wild-type*)
                    (sb-kernel:type-specifier (sb-kernel::array-type-specialized-element-type type)))))
        (cond ((let ((type (element-type type)))
                 (and type
                      `',type)))
              ((sb-kernel:union-type-p type)
               (let (result)
                 (loop for type in (sb-kernel:union-type-types type)
                       for et = (element-type type)
                       unless (and et
                                   (if result
                                       (equal result et)
                                       (setf result et)))
                       do (sb-c::give-up-ir1-transform))
                 `',result))
              ((sb-kernel:intersection-type-p type)
               (loop for type in (sb-kernel:intersection-type-types type)
                     for et = (element-type type)
                     when et
                     return `',et
                     finally (sb-c::give-up-ir1-transform)))
              (t
               (sb-c::give-up-ir1-transform)))))))

;;;
;;; Arithmetic operations with dynamic mod
;;;

(declaim ((unsigned-byte 32) *modulus*))
(defvar *modulus*)

(defun mod* (&rest args)
  (reduce (lambda (x y) (mod (* x y) *modulus*)) args))

(define-compiler-macro mod* (&rest args)
  (if (null args)
      1
      (reduce (lambda (x y) `(mod (* ,x ,y) *modulus*)) args)))

(defun mod+ (&rest args)
  (reduce (lambda (x y) (mod (+ x y) *modulus*)) args))

(define-compiler-macro mod+ (&rest args)
  (if (null args)
      0
      (reduce (lambda (x y) `(mod (+ ,x ,y) *modulus*)) args)))

(define-modify-macro incfmod (delta &optional (divisor '*modulus*))
  (lambda (x y divisor) (mod (+ x y) divisor)))

(define-modify-macro decfmod (delta &optional (divisor '*modulus*))
  (lambda (x y divisor) (mod (- x y) divisor)))


;;
;; Matrix multiplication over semiring
;;

(declaim (inline gemm))
(defun gemm (a b &key (op+ #'+) (op* #'*) (identity+ 0))
  "Calculates A*B. (OP+, OP*) must form a semiring. IDENTITY+ is the identity
element w.r.t. OP+."
  (declare ((simple-array * (* *)) a b)
           (function op+ op*))
  (let ((c (make-array (list (array-dimension a 0) (array-dimension b 1))
                       :element-type (array-element-type a))))
    (dotimes (row (array-dimension a 0))
      (dotimes (col (array-dimension b 1))
        (let ((res identity+))
          (dotimes (k (array-dimension a 1))
            (setf res
                  (funcall op+ res (funcall op* (aref a row k) (aref b k col)))))
          (setf (aref c row col) res))))
    c))

(declaim (inline matrix-power))
(defun matrix-power (base power &key (op+ #'+) (op* #'*) (identity+ 0) (identity* 1))
  (declare ((simple-array * (* *)) base)
           (function op+ op*)
           ((integer 0 #.most-positive-fixnum) power))
  (let ((size (array-dimension base 0)))
    (assert (= size (array-dimension base 1)))
    (let ((iden (make-array (array-dimensions base)
                            :element-type (array-element-type base)
                            :initial-element identity+)))
      (dotimes (i size)
        (setf (aref iden i i) identity*))
      (labels ((recur (p)
                 (declare ((integer 0 #.most-positive-fixnum) p))
                 (cond ((zerop p) iden)
                       ((evenp p)
                        (let ((res (recur (ash p -1))))
                          (gemm res res :op+ op+ :op* op* :identity+ identity+)))
                       (t
                        (gemm base (recur (- p 1))
                              :op+ op+ :op* op* :identity+ identity+)))))
        (recur power)))))

(declaim (inline gemv))
(defun gemv (a x &key (op+ #'+) (op* #'*) (identity+ 0))
  "Calculates A*x for a matrix A and a vector x. (OP+, OP*) must form a
semiring. IDENTITY+ is the identity element w.r.t. OP+."
  (declare ((simple-array * (* *)) a)
           ((simple-array * (*)) x)
           (function op+ op*))
  (let ((y (make-array (array-dimension a 0) :element-type (array-element-type x))))
    (dotimes (i (length y))
      (let ((res identity+))
        (dotimes (j (length x))
          (setf res
                (funcall op+ res (funcall op* (aref a i j) (aref x j)))))
        (setf (aref y i) res)))
    y))

(declaim (inline power-mod))
(defun power-mod (base power &optional (divisor 1000000007))
  "BASE := integer
POWER, DIVISOR := non-negative fixnum"
  (declare ((integer 0 #.most-positive-fixnum) divisor)
           (integer base))
  (labels ((recur (x p)
             (declare ((integer 0 #.most-positive-fixnum) x p))
             (cond ((zerop p) 1)
                   ((evenp p) (recur (mod (* x x) divisor) (ash p -1)))
                   (t (mod (* x (recur x (- p 1))) divisor)))))
    (declare (ftype (function * (values (integer 0 #.most-positive-fixnum) &optional))
                    recur))
    (recur (mod base divisor) power)))

(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)))

(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)

(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 main ()
  (let* ((l (read))
         (a (read))
         (b (read))
         (m (read))
         (boundaries (make-array 19 :element-type 'uint62 :initial-element 0)))
    (declare (uint62 l a b)
             (uint31 m)
             ((simple-array uint62 (*)) boundaries))
    (let ((*modulus* m))
      (dotimes (d 19)
        (setf (aref boundaries d)
              (max 0 (ceiling (- (expt 10 d) a) b))))
      (dotimes (d 19)
        (when (>= (aref boundaries d) l)
          (setf (aref boundaries d) l)
          (setf boundaries (adjust-array boundaries (+ d 1)))
          (return)))
      (let ((res 0))
        (declare (uint31 res))
        (loop for d from 1 below (length boundaries)
              for length = (- (aref boundaries d) (aref boundaries (- d 1)))
              for vec = (make-array 3 :element-type 'uint32)
              for mat = (make-array (list 3 3) :element-type 'uint32 :initial-element 0)
              do (setf (aref vec 0) 0
                       (aref vec 1) 0
                       (aref vec 2) 1)
                 (setf (aref mat 0 0) (mod (expt 10 d) m)
                       (aref mat 0 1) (mod b m)
                       (aref mat 0 2) (mod+ a (mod* b (aref boundaries (- d 1))))
                       (aref mat 1 1) 1
                       (aref mat 1 2) 1
                       (aref mat 2 2) 1)
                 (setq vec (gemv (matrix-power mat length
                                               :op+ #'mod+
                                               :op* #'mod*)
                                 vec
                                 :op+ #'mod+
                                 :op* #'mod*))
                 (setf res
                       (mod+ (mod* res (power-mod (expt 10 d) length m))
                             (aref vec 0))))
        (println res)))))

#-swank(main)