Question 56
On the circumference of the circle with radius $1$, points $A_1$, $A_2$, $A_3$, $A_4$, $A_5$ are placed in this order, and they satisfy
$$
\angle A_5A_2A_4 = \angle A_1A_3A_5 = \angle A_2A_4A_1 = \angle A_3A_5A_2 = 30^{\circ}.
$$
Let $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ be the points of intersections of $A_2A_4$ and $A_3A_5$, $A_3A_5$ and $A_4A_1$, $A_4A_1$ and $A_5A_2$, $A_5A_2$ and $A_1A_3$, $A_1A_3$ and $A_2A_4$, respectively. Find the area of the pentagon $B_1B_2B_3B_4B_5$.
$$
\angle A_5A_2A_4 = \angle A_1A_3A_5 = \angle A_2A_4A_1 = \angle A_3A_5A_2 = 30^{\circ}.
$$
Let $B_1$, $B_2$, $B_3$, $B_4$, $B_5$ be the points of intersections of $A_2A_4$ and $A_3A_5$, $A_3A_5$ and $A_4A_1$, $A_4A_1$ and $A_5A_2$, $A_5A_2$ and $A_1A_3$, $A_1A_3$ and $A_2A_4$, respectively. Find the area of the pentagon $B_1B_2B_3B_4B_5$.