Find the area!

Geometry Level 5

$S=\{(x,y,z) \in \mathbb{R}^3: x^2+y^2+z^2=1\}$

Let $$S$$ be the surface of a unit sphere centered at the origin.

Let $$F$$ be a subset of $$S$$. Suppose that there exist three subsets $$F_1,F_2,F_3$$ of $$S$$, each of which is obtained by rigidly rotating the set $$F$$ around the origin such that:

1. These three sets are pairwise disjoint, i.e., $F_i\cap F_j = \phi, i\neq j$
2. Their union spans the entire surface of the sphere, i.e.,
$F_1\cup F_2 \cup F_3=S$

Find the area of any such set $$F$$ satisfying the above conditions.

P.S. Rotation is defined in the usual way: $F_i=\{(x,y,z): \exists (a,b,c) \in F \text{ s.t. } (x,y,z)'=Q_i (a,b,c)'\}$ for some orthonormal matrix $$Q_i,i=1,2,3$$.

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