Three Countries

Today, you want to measure the accessible area of Teyvat.

Mondstadt, Liyue, and Inazuma are the three countries in Teyvat. The territories of these countries can be regarded as three circles $c\_1$, $c\_2$, and $c\_3$, respectively. It is possible that some of the circles overlap.

Let $S\_i$ be the set of points in $c\_i$. The area of Teyvat, $S$, is defined as the convex hull of points in $S\_1 \cup S\_2 \cup S\_3$.

Formally, $S$ is the smallest set of points satisfying the following two conditions:

• $S \supseteq S\_1 \cup S\_2 \cup S\_3$,
• $\forall p\_1, p\_2 \in S, \forall \alpha \in \[0, 1], \alpha p\_1 + (1-\alpha)p\_2 \in S$.

You are given the circles $c\_1$, $c\_2$, and $c\_3$. Your task is to calculate the area of $S$.

The first line contains a single integer $t$, the number of test cases ($1 \leq t \leq 10^4$).

Each test case is given on three lines. The $i$-th of these lines contains three integers, $x$, $y$, and $r$, which are the coordinates of the center and the radius of $i$-th circle ($1 \leq x, y ,r \leq 100$).

For each test case, output a single real number representing the area of $S$. 

Your answer will be considered correct if its absolute or relative error when compared with the jury's answer is no more than $10^{-6}$.