USAMO 2021 Problem 1

Rectangles $BCC_1B_2,$ $CAA_1C_2,$ and $ABB_1A_2$ are erected outside an acute triangle $ABC.$ Suppose that $$\angle BC_1C+\angle CA_1A+\angle AB_1B=180^{\circ}.$$ Prove that lines $B_1C_2,$ $C_1A_2,$ and $A_1B_2$ are concurrent.