Each side of $1 \times 1$ square is a hypothenuse of an exterior right triangle. Let $A, B, C, D$ be the vertices of the right angles and $O_1, O_2, O_3, O_4$ be the centers of the incircles of these triangles. Prove that $a)$ The area of quadrilateral $ABCD$ does not exceed $2$; $b)$ The area of quadrilateral $O_1O_2O_3O_4$ does not exceed $1$.