8621: Black Hole

Twilight Sparkle finds that each of the black holes is a convex regular polyhedron with $n$ faces (a convex polyhedron whose $n$ faces are identical regular polygons), and the length of its edges is $a$.

Every day the black hole will shrink exactly once. If a black hole shrinks, it will become the convex hull of $n$ geometry centers of the original black hole's $n$ faces. And whenever the black hole isn't a regular polyhedron, it will disappear forever.

Twilight Sparkle has made $T$ records for the black holes she has observed. Each of the records can be described as a tuple (n,a,k)(n,a,k)(n,a,k), indicating that a convex regular polyhedron black hole with edges of length $a$ and $n$ faces initially has shrunk $k$ times. However, naughty Rainbow Dash replaced some of the records with impossible ones. Can you find out the impossible records and calculate the final n′n'n′ and a′a'a′ after $k$ times of shrinking for those possible records?

The first line contains an integer $T$ (1≤T≤100)(1\le T\le 100)(1≤T≤100), denoting the number of the records.
In the following $T$ lines, each line describes a record and contains three integers n,a,kn,a,kn,a,k ($1\le n\le 200$, $1\le a\le 1000$, 1≤k≤201\le k\le 201≤k≤20).

For each of the records, if the record is impossible, output $\text{impossible}$ in a single line. Otherwise, output $\text{possible}$ followed by an integer n′n'n′ and a real number a′a'a′ in a single line, separated by spaces. a′a'a′ is considered correct if the relative or absolute error between yours and the standard solution is not greater than 10−610^{-6}10−6.

3
2 10 1
4 10 1
6 10 1

impossible
possible 4 3.333333333333
possible 8 7.071067811865

For the first record, it's impossible to find a convex regular polyhedron with $2$ faces.
For the second record, a $4$-face convex regular polyhedron is a tetrahedron. After taking all its faces' geometry centers to make a convex hull, we get a smaller tetrahedron.
For the third record, a $6$-face convex regular polyhedron is a cube. After taking all its faces' geometry centers to make a convex hull, we get an octahedron, a.k.a. an $8$-face convex regular polyhedron.