1992 AIME Problem 11

Lines l_1^{} and l_2^{} both pass through the origin and make first-quadrant angles of \frac{\pi}{70} and \frac{\pi}{54} radians, respectively, with the positive x-axis. For any line l^{}_{}, the transformation R(l)^{}_{} produces another line as follows: l^{}_{} is reflected in l_1^{}, and the resulting line is reflected in l_2^{}. Let R^{(1)}(l)=R(l)^{}_{} and R^{(n)}(l)^{}_{}=R\left(R^{(n-1)}(l)\right). Given that l^{}_{} is the line y=\frac{19}{92}x^{}_{}, find the smallest positive integer m^{}_{} for which R^{(m)}(l)=l^{}_{}.

Leading zeroes must be inputted, so if your answer is 34, then input 034

Full credit goes to MAA for authoring these problems. These problems were taken on the AOPS website.