7046：Fall with Trees

Fall wants to draw a perfect binary tree.

We first stipulate that all nodes in the tree with the same depth also have the same y-coordinate in the plane. Define nodes with the same depth to be nodes at the same level, then the perfect binary tree has four properties.

- It is a full binary tree.
- The difference between the y-coordinates of two nodes at each adjacent level is a constant.
- The difference between the x-coordinates of two adjacent nodes at the same level is constant.
- The x-coordinate of each node is the average of the x-coordinates of its sons.

Fall has drawn the root node and its left and right sons of this binary tree. Now Fall intends to draw a total of $k$ levels and cut the binary tree down and paste it on the wall afterwards, so he wants to know what is the area of the convex hull of all nodes of this perfect binary tree.

The input consists of multiple test cases.

The first line contains an integer $T$ ($T \leq 2\times 10^5$) -- the number of test cases.

For each test case:

In the first line, there is an integer $k$ ($2 \leq k \leq 10^4$).

In the second line, there are six integers $x_{root},y_{root},x_{lson},y_{lson},x_{rson},y_{rson}\in [-10^4,10^4]$ ,which represent the coordinates of the root node and its sons.

It is guaranteed that all the coordinates meet the conditions of the question, which means:
- $x_{lson}+x_{rson}=2\times x_{root}$
- $y_{lson}=y_{rson}$
- $y_{root}>y_{lson},x_{lson}<x_{rson}$

For each test case, output a real number representing the answer, with three decimal places.