SHARYGIN 2012 Problem 6

Point $C_1$ of hypothenuse $AC$ of a right-angled triangle $ABC$ is such that $BC = CC_1$. Point $C_2$ on cathetus $AB$ is such that $AC_2 = AC_1$; point $A_2$ is defined similarly. Find angle $AMC$, where $M$ is the midpoint of $A_2C_2$.