HMMT 2024 G10

Suppose point $P$ is inside quadrilateral $A B C D$ such that

$$\begin{aligned} & \angle P A B=\angle P D A \\ & \angle P A D=\angle P D C, \\ & \angle P B A=\angle P C B, \text { and } \\ & \angle P B C=\angle P C D \end{aligned}$$

If $P A=4, P B=5$, and $P C=10$, compute the perimeter of $A B C D$.