## ASUMEXTR - A Summatory (Extreme)

f(n) is defined as: f(n) = 1k+2k+3k+...+nk, so it is the sum of the k-th power of all natural numbers up to n.

In this problem you are about to compute,

f(1) + f(2) + f(3) + ... + f(n)

Note: This is a harder version of the problem ASUMHARD, with larger constraints. Please read the constraints section carefully.

### Input

The first line is an integer T, denoting the number of test cases. Then, T test cases follow.

For each test case, there are two integers n and k written in one line, separated by space.

### Output

For each test case, output the result of the summatory function described above.

Since this number could be very large, output the answer modulo 1,234,567,891.

### Example

```Input:
5
2 3
10 3
3 3
100 0
100 1

Output:
10
7942
46
5050
171700```

### Explanation

In case 1, n = 2, k = 3. f(1) = 13, f(2) = 13+23. ans = f(1) + f(2) = 10.

### Constraints

Overall constraints

• 5 ≤ T ≤ 500000
• 1 ≤ n ≤ 1018
• 0 ≤ k ≤ 10000000

More precise information (there are 6 test files):

Test #0: T = 500000, 0 ≤ k ≤ 100

Test #1: T = 50000, 0 ≤ k ≤ 1000

Test #2: T = 5000, 0 ≤ k ≤ 10000

Test #3: T = 500, 0 ≤ k ≤ 100000

Test #4: T = 50, 0 ≤ k ≤ 1000000

Test #5: T = 5, 0 ≤ k ≤ 10000000

It should be clear from the constraints that an O(k2) solution will not pass. Inputs are generated uniformly randomly in the given ranges (with some manual worst case inputs). Time limit is set to 2x of my unoptimized C++ code.

 Added by: suhash Date: 2020-08-17 Time limit: 20s Source limit: 50000B Memory limit: 1536MB Cluster: Cube (Intel G860) Languages: All Resource: ASUMHARD