CZ-SK-MO 2011 Problem 1

In a certain triangle $ABC$ , there are points $K$ and $M$ on sides $AB$ and $AC$ , respectively, such that if $L$ is the intersection of $MB$ and $KC$ , then both $AKLM$ and $KBCM$ are cyclic quadrilaterals with the same size circumcircles. Find the measures of the interior angles of triangle $ABC$ .