Ntarsis has a box $$$B$$$ with side lengths $$$x$$$, $$$y$$$, and $$$z$$$. It lies in the 3D coordinate plane, extending from $$$(0,0,0)$$$ to $$$(x,y,z)$$$.

Ntarsis has a secret box $$$S$$$. He wants to choose its dimensions such that all side lengths are positive integers, and the volume of $$$S$$$ is $$$k$$$. He can place $$$S$$$ somewhere within $$$B$$$ such that:

$$$S$$$ is parallel to all axes.
every corner of $$$S$$$ lies on an integer coordinate.

$$$S$$$ is magical, so when placed at an integer location inside $$$B$$$, it will not fall to the ground.

Among all possible ways to choose the dimensions of $$$S$$$, determine the maximum number of distinct locations he can choose to place his secret box $$$S$$$ inside $$$B$$$. Ntarsis does not rotate $$$S$$$ once its side lengths are selected.

Input

The first line consists of an integer $$$t$$$, the number of test cases ($$$1 \leq t \leq 2000$$$). The description of the test cases follows.

The first and only line of each test case contains four integers $$$x, y, z$$$ and $$$k$$$ ($$$1 \leq x, y, z \leq 2000$$$, $$$1 \leq k \leq x \cdot y \cdot z$$$).

It is guaranteed the sum of all $$$x$$$, sum of all $$$y$$$, and sum of all $$$z$$$ do not exceed $$$2000$$$ over all test cases.

Note that $$$k$$$ may not fit in a standard 32-bit integer data type.

Output

For each test case, output the answer as an integer on a new line. If there is no way to select the dimensions of $$$S$$$ so it fits in $$$B$$$, output $$$0$$$.

Note

For the first test case, it is optimal to choose $$$S$$$ with side lengths $$$2$$$, $$$2$$$, and $$$2$$$, which has a volume of $$$2 \cdot 2 \cdot 2 = 8$$$. It can be shown there are $$$8$$$ ways to put $$$S$$$ inside $$$B$$$.

The coordinate with the least $$$x$$$, $$$y$$$, and $$$z$$$ values for each possible arrangement of $$$S$$$ are:

$$$(0, 0, 0)$$$
$$$(1, 0, 0)$$$
$$$(0, 1, 0)$$$
$$$(0, 0, 1)$$$
$$$(1, 0, 1)$$$
$$$(1, 1, 0)$$$
$$$(0, 1, 1)$$$
$$$(1, 1, 1)$$$

The arrangement of $$$S$$$ with a coordinate of $$$(0, 0, 0)$$$ is depicted below:

$\text{[math]}$

For the second test case, $$$S$$$ with side lengths $$$2$$$, $$$3$$$, and $$$3$$$ are optimal.

Secret Box

Examples