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 C# (Mono 4.6.2.0) (judge verdict: Time Limit Exceeded)

using System;
using System.Collections.Generic;
using System.Diagnostics;
using System.Linq;
using System.Text;
using static System.Math;

class Program
{
    static long mod = (long)1e6 + 3;
    static long[] f = new long[mod]; //階乗
    static long[] invf = new long[mod]; //階乗の逆元

    static void Main()
    {
        f[0] = 1;
        invf[0] = 1;
        for (int i = 1; i < mod; i++)
        {
            f[i] = f[i - 1] * i % mod;
            invf[i] = inv(f[i]);
        }
        int q = int.Parse(Console.ReadLine());
        for (int i = 0; i < q; i++)
        {
            long[] input = Console.ReadLine().Split().Select(long.Parse).ToArray();
            Console.WriteLine(Ans(input[0], input[1], input[2]));
        }
    }

    static long Ans(long x, long d, long n)
    {
        for (long k = 1; k <= (x + (n - 1) * d) / mod; k++)
        {
            if ((k * mod - x) % d == 0 && 0 <= (k * mod - x) / d && (k * mod - x) / d < n)
                return 0;
        }
        long y = x * inv(d) % mod;
        long r = (y + n - 1 + mod) % mod;
        long l = (y - 1 + mod) % mod;
        long ans = pow(d, n);
        return ans * f[r] % mod * invf[l] % mod;
    }

    static long pow(long a, long k)
    {
        if (k == 0) return 1;
        if (k == 1) return a % mod;
        long t = pow(a, k / 2);
        if (k % 2 == 0) return t * t % mod;
        return t * t % mod * a % mod;
    }

    static long inv(long k)
    {
        return pow(k, mod - 2);
    }
}