F. Lighthouse
time limit per test
2 seconds
memory limit per test
256 megabytes
input
standard input
output
standard output

A lighthouse stands vertically at sea level.

An observer is standing on a ship at an unknown eye height above sea level.

The total height of the lighthouse is $$$H$$$ meters.

The angle of depression from the observer to the base of the lighthouse is $$$\alpha$$$ degrees, and the angle of elevation from the observer to the top of the lighthouse is $$$\beta$$$ degrees.

This figure illustrates the first sample. Here, $$$H = 60$$$, $$$\alpha = 30^\circ$$$ is the angle of depression, and $$$\beta = 60^\circ$$$ is the angle of elevation.

For each test case, determine the observer's eye height above sea level.

Input

The first line contains one integer $$$t$$$ $$$(1 \le t \le 2 \cdot 10^5)$$$ — the number of test cases.

Each of the next $$$t$$$ lines contains three integers $$$\alpha, \beta, H$$$ where:

  • $$$1 \le \alpha \le 89$$$ — the angle of depression, measured in degrees.
  • $$$1 \le \beta \le 89$$$ — the angle of elevation, measured in degrees.
  • $$$1 \le H \le 10^6$$$ — the total height of the lighthouse, measured in meters.

It is guaranteed that all test cases are valid.

Output

For each test case, print the observer's eye height above sea level.

Your answer will be considered correct if its absolute or relative error does not exceed $$$10^{-6}$$$.

Example
Input
3
30 60 60
45 45 20
45 45 40
Output
15.0000000000
10.0000000000
20.0000000000
Note

The angle of depression is measured from the horizontal line passing through the observer's eye level down to the line of sight toward the base of the lighthouse, while the angle of elevation is measured from the horizontal line passing through the observer's eye level up to the line of sight toward the top of the lighthouse.