IZhO 2008 Problem 5

Let $A_1A_2$ be the external tangent line to the nonintersecting cirlces $\omega_1(O_1)$ and $\omega_2(O_2)$ , $A_1\in\omega_1$ , $A_2\in\omega_2$ .Points $K$ is the midpoint of $A_1A_2$ .And $KB_1$ and $KB_2$ are tangent lines to $\omega_1$ and $\omega_2$ ,respectvely( $B_1\neq A_1$ , $B_2\neq A_2$ ).Lines $A_1B_1$ and $A_2B_2$ meet in point $L$ ,and lines $KL$ and $O_1O_2$ meet in point $P$ . Prove that points $B_1,B_2,P$ and $L$ are concyclic.