Let $ABC$ be an acute triangle. Let $D$ be the reflection of point $B$ with respect to the line $AC$. Let $E$ be the reflection of point $C$ with respect to the line $AB$. Let $\Gamma_1$ be the circle that passes through $A, B$, and $D$. Let $\Gamma_2$ be the circle that passes through $A, C$, and $E$. Let $P$ be the intersection of $\Gamma_1$ and $\Gamma_2$ , other than $A$. Let $\Gamma$ be the circle that passes through $A, B$, and $C$. Show that the center of $\Gamma$ lies on line $AP$.