SHARYGIN 2020 Problem 10

Given are a closed broken line $A_1A_2\ldots A_n$ and a circle $\omega$ which touches each of lines $A_1A_2,A_2A_3,\ldots,A_nA_1$ . Call the link good, if it touches $\omega$ , and bad otherwise (i.e. if the extension of this link touches $\omega$ ). Prove that the number of bad links is even.