Problem:
Score : 600 points

Problem Statement

Consider the following arithmetic progression with n terms:

x, x + d, x + 2d, \ldots, x + (n-1)d

What is the product of all terms in this sequence?
Compute the answer modulo 1\ 000\ 003.

You are given Q queries of this form.
In the i-th query, compute the answer in case x = x_i, d = d_i, n = n_i.

Constraints

1 \leq Q \leq 10^5

0 \leq x_i, d_i \leq 1\ 000\ 002

1 \leq n_i \leq 10^9

All values in input are integers.

Input

Input is given from Standard Input in the following format:

Q
x_1 d_1 n_1
:
x_Q d_Q n_Q

Output

Print Q lines.

In the i-th line, print the answer for the i-th query.

Sample Input 1

2
7 2 4
12345 67890 2019

Sample Output 1

9009
916936

For the first query, the answer is 7 \times 9 \times 11 \times 13 = 9009.
Don't forget to compute the answer modulo 1\ 000\ 003.

Solution in Haskell (GHC 7.10.3) (judge verdict: Accepted)

{-# OPTIONS_GHC -O2 #-}
{-# LANGUAGE BangPatterns          #-}
{-# LANGUAGE CPP                   #-}
{-# LANGUAGE LambdaCase            #-}
{-# LANGUAGE MagicHash             #-}
{-# LANGUAGE MultiParamTypeClasses #-}
{-# LANGUAGE MultiWayIf            #-}
{-# LANGUAGE OverloadedStrings     #-}
{-# LANGUAGE TupleSections         #-}
{-# LANGUAGE TypeFamilies          #-}
{-# LANGUAGE ViewPatterns          #-}

import           Control.Applicative
import           Control.Exception
import           Control.Monad
import           Control.Monad.Primitive
import           Control.Monad.ST
import           Control.Monad.State.Strict
import           Data.Bool
import qualified Data.ByteString             as B
import qualified Data.ByteString.Char8       as C
import qualified Data.ByteString.Unsafe      as B
import           Data.Char
import qualified Data.Foldable               as F
import           Data.Function
import qualified Data.IntMap.Strict          as IM
import qualified Data.IntSet                 as IS
import qualified Data.List                   as L
import qualified Data.Map.Strict             as M
import           Data.Monoid
import           Data.Ord
import           Data.Primitive.MutVar
import qualified Data.Set                    as S
import           Data.Tuple
import qualified Data.Vector                 as V
import qualified Data.Vector.Generic         as G
import qualified Data.Vector.Generic.Mutable as GM
import qualified Data.Vector.Unboxed         as U
import qualified Data.Vector.Unboxed.Mutable as UM
import           Debug.Trace
import           Foreign                     hiding (void)
import           GHC.Exts
import           Unsafe.Coerce

main :: IO ()
main = do
    !n <- readLn :: IO Int
    qs <- U.unfoldrN n parseInt3 <$> C.getContents
    putStr.unlines.map show.U.toList $ solve n qs

solve :: Int -> U.Vector (Int, Int, Int) -> U.Vector Int
solve _ = U.map solve'

solve' :: (Int, Int, Int) -> Int
solve' (x, d, n)
    | d == 0 = x ^% n
    | n >= modulus = 0
    | otherwise = (d ^% n) *% (fact (xd + n - 1) /% fact (xd - 1))
  where
    !xd = x /% d

-------------------------------------------------------------------------------
#define MOD 1000003

modulus :: Int
modulus = MOD

infixr 8 ^%
infixl 7 *%, /%
infixl 6 +%, -%

type IntMod = Int

intMod :: Int -> IntMod
intMod x = rem x MOD

intModValidate :: Int -> Bool
intModValidate x = 0 <= x && x < MOD

(+%) :: IntMod -> IntMod -> IntMod
(I# x#) +% (I# y#) = case x# +# y# of
    r# -> I# (r# -# ((r# >=# MOD#) *# MOD#))
{-# INLINE (+%) #-}

(-%) :: IntMod -> IntMod -> IntMod
(I# x#) -% (I# y#) = case x# -# y# of
    r# -> I# (r# +# ((r# <# 0#) *# MOD#))
{-# INLINE (-%) #-}

(*%) :: IntMod -> IntMod -> IntMod
(I# x#) *% (I# y#) = I# ((x# *# y#) `remInt#` MOD#)
{-# INLINE (*%) #-}

(/%) :: IntMod -> IntMod -> IntMod
(I# x#) /% (I# y#) = go# y# MOD# 1# 0#
  where
    go# a# b# u# v#
        | isTrue# (b# ># 0#) = case a# `quotInt#` b# of
            q# -> go# b# (a# -# (q# *# b#)) v# (u# -# (q# *# v#))
        | otherwise = I# ((x# *# (u# +# MOD#)) `remInt#` MOD#)
{-# INLINE (/%) #-}

(^%) :: IntMod -> Int -> IntMod
x ^% n
    | n > 0 = go 1 x n
    | n == 0 = 1
    | otherwise = go 1 (1 /% x) (-n)
  where
    go !acc !y !m
        | m .&. 1 == 0 = go acc (y *% y) (unsafeShiftR m 1)
        | m == 1 = acc *% y
        | otherwise = go (acc *% y) (y *% y) (unsafeShiftR (m - 1) 1)

fact :: Int -> IntMod
fact = U.unsafeIndex factCache
{-# INLINE fact #-}

recipFact :: Int -> IntMod
recipFact = U.unsafeIndex recipFactCache
{-# INLINE recipFact #-}

comb :: Int -> Int -> IntMod
comb n k = fact n *% recipFact (n - k) *% recipFact k
{-# INLINE comb #-}

factCacheSize :: Int
factCacheSize = 2000200
{-# INLINE factCacheSize #-}

factCache :: U.Vector IntMod
factCache = U.scanl' (*%) 1 $ U.generate factCacheSize (+1)
{-# NOINLINE factCache #-}

recipFactCache :: U.Vector IntMod
recipFactCache = U.scanr' (*%) (1 /% (factCache U.! factCacheSize))
    $ U.generate factCacheSize (+1)
{-# NOINLINE recipFactCache #-}

type Parser a = C.ByteString -> Maybe (a, C.ByteString)

parseInt :: Parser Int
parseInt = C.readInt . C.dropWhile isSpace

parseInt2 :: Parser (Int, Int)
parseInt2 = runStateT $
    (,) <$> StateT (C.readInt . C.dropWhile isSpace)
        <*> StateT (C.readInt . B.unsafeTail)

parseInt3 :: Parser (Int, Int, Int)
parseInt3 = runStateT $
    (,,) <$> StateT (C.readInt . C.dropWhile isSpace)
        <*> StateT (C.readInt . B.unsafeTail)
        <*> StateT (C.readInt . B.unsafeTail)

parseInt4 :: Parser (Int, Int, Int, Int)
parseInt4 = runStateT $
    (,,,) <$> StateT (C.readInt . C.dropWhile isSpace)
        <*> StateT (C.readInt . B.unsafeTail)
        <*> StateT (C.readInt . B.unsafeTail)
        <*> StateT (C.readInt . B.unsafeTail)