$A_1A_2A_3A_4$ is a cyclic quadrilateral inscribed in circle $\Omega$, with side lengths $A_1A_2 = 28$, $A_2A_3 =12\sqrt3$, $A_3A_4 = 28\sqrt3$, and $A_4A_1 = 8$. Let $X$ be the intersection of $A_1A_3, A_2A_4$. Now, for $i = 1, 2, 3, 4$, let $\omega_i$ be the circle tangent to segments$ A_iX$, $A_{i+1}X$, and $\Omega$, where we take indices cyclically (mod $4$). Furthermore, for each $i$, say $\omega_i$ is tangent to $A_1A_3$ at $X_i $, $A_2A_4$ at $Y_i$ , and $\Omega$ at $T_i$ . Let $P_1$ be the intersection of $T_1X_1$ and $T_2X_2$, and $P_3$ the intersection of $T_3X_3$ and $T_4X_4$. Let $P_2$ be the intersection of $T_2Y_2$ and $T_3Y_3$, and $P_4$ the intersection of $T_1Y_1$ and $T_4Y_4$. Find the area of quadrilateral $P_1P_2P_3P_4$.

Let's think step by step and output the final answer within \boxed{}.
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