A11355. Paper Cutting Again

There is a rectangular sheet of paper with initial height $n$ and width $m$ . Let the current height and width be $h$ and $w$ respectively. We introduce a $xy$ -coordinate system so that the four corners of the sheet are $(0, 0), (w, 0), (0, h)$ , and $(w, h)$ . The sheet can then be cut along the lines $x = 1,2,\ldots,w-1$ and the lines $y = 1,2,\ldots,h-1$ . In each step, the paper is cut randomly along any one of these $h+w-2$ lines. After each vertical and horizontal cut, the right and bottom piece of paper respectively are discarded.

Find the expected number of steps required to make the area of the sheet of paper strictly less than $k$ . It can be shown that this answer can always be expressed as a fraction $\dfrac{p}{q}$ where $p$ and $q$ are coprime integers. Calculate $p\cdot q^{-1} \bmod (10^9+7)$ .

Each test contains multiple test cases. The first line contains the number of test cases $t$ ( $1 \le t \le 57000$ ). Description of the test cases follows.

The first line of each test case contains 3 integers $n$ , $m$ , and $k$ ( $1 \le n, m \le 10^6$ , $2 \le k \le 10^{12}$ ).

It is guaranteed that the sum of $n$ and the sum of $m$ over all test cases do not exceed $10^6$ .

For each test case, print one integer — the answer to the problem.

For the first test case, the area is already less than $10$ so no cuts are required.

For the second test case, the area is exactly $8$ so any one of the $4$ possible cuts would make the area strictly less than $8$ .

For the third test case, the final answer is $\frac{17}{6} = 833\,333\,342\bmod (10^9+7)$ .

For the fourth test case, the final answer is $\frac{5}{4} = 250\,000\,003\bmod (10^9+7)$ .