F. Hamed and AghaBalaSar
time limit per test
2 seconds
memory limit per test
512 megabytes
input
standard input
output
standard output

Hamid wrote $$$3014$$$ lines of code for the a+b problem and gave $$$10$$$ lines of it to Hamed; then the following problem came to Hamed's mind.

You are given two integers $$$n$$$ and $$$m$$$.

An array $$$a_1,a_2,\ldots,a_n$$$ is called snake if and only if all of the following conditions hold:

  • All elements in $$$a$$$ are integers between $$$0$$$ and $$$m$$$;
  • $$$a_1 + a_2 + \cdots + a_n = m$$$;
  • $$$a_n=\max\left([a_1, a_2, \ldots, a_n]\right)$$$.

We define $$$f(a)$$$ as in the following pseudocode:

function f(array a):
    pos := 1
    res := 0
    let nxt[x] be an array such that nxt[x] is the smallest index y such that
                                    y > x and a[y] > a[x],
                              or undefined if no such y exists
    while pos < n:
        if a[pos] < a[n]:
            res += a[nxt[pos]] - a[pos]
            pos := nxt[pos]
        else:
            pos += 1
    return res

Find the sum of $$$f(a)$$$ over all snake arrays $$$a_1,a_2,\ldots,a_n$$$, modulo $$$10^9+7$$$.

Input

Each test contains multiple test cases. The first line contains the number of test cases $$$t$$$ ($$$1 \le t \le 10^4$$$). The description of the test cases follows.

The first line of each test case contains two integers $$$n$$$ and $$$m$$$ ($$$2 \leq n \leq 2 \cdot 10^5$$$, $$$0 \leq m \leq 2 \cdot 10^5$$$) — the length of a snake array and the sum of all elements in a snake array.

It is guaranteed that the sum of $$$m$$$ over all test cases does not exceed $$$2 \cdot 10^5$$$.

Output

For each test case, output a single integer — the sum of $$$f(a)$$$ over all snake arrays $$$a_1,a_2,\ldots,a_n$$$, modulo $$$10^9+7$$$.

Example
Input
8
2 5
3 4
4 6
5 10
6 23
100 100
142857 33333
200000 0
Output
9
14
76
985
142112
771227753
865580631
0
Note

In the first test case, there are three snake arrays:

  • $$$f([0, 5]) = 5$$$;
  • $$$f([1, 4]) = 3$$$;
  • $$$f([2, 3]) = 1$$$.

Thus, the answer is $$$5+3+1=9$$$.

In the second test case, there are six snake arrays:

  • $$$f([0, 0, 4]) = 4$$$;
  • $$$f([0, 1, 3]) = 3$$$;
  • $$$f([1, 0, 3]) = 2$$$;
  • $$$f([1, 1, 2]) = 1$$$;
  • $$$f([0, 2, 2]) = 2$$$;
  • $$$f([2, 0, 2]) = 2$$$.

Thus, the answer is $$$4+3+2+1+2+2=14$$$.

In the fifth test case, a possible snake array is:

  • $$$f([3, 1, 4, 1, 5, 9]) = 6$$$.