CF1354C2.Not So Simple Polygon Embedding

The statement of this problem is the same as the statement of problem C1. The only difference is that, in problem C1, $n$ is always even, and in C2, $n$ is always odd.

You are given a regular polygon with $2 \cdot n$ vertices (it's convex and has equal sides and equal angles) and all its sides have length $1$ . Let's name it as $2n$ -gon.

Your task is to find the square of the minimum size such that you can embed $2n$ -gon in the square. Embedding $2n$ -gon in the square means that you need to place $2n$ -gon in the square in such way that each point which lies inside or on a border of $2n$ -gon should also lie inside or on a border of the square.

You can rotate $2n$ -gon and/or the square.

The first line contains a single integer $T$ ( $1 \le T \le 200$ ) — the number of test cases.

Next $T$ lines contain descriptions of test cases — one per line. Each line contains single odd integer $n$ ( $3 \le n \le 199$ ). Don't forget you need to embed $2n$ -gon, not an $n$ -gon.

Print $T$ real numbers — one per test case. For each test case, print the minimum length of a side of the square $2n$ -gon can be embedded in. Your answer will be considered correct if its absolute or relative error doesn't exceed $10^{-6}$ .