1992 AIME Problem 11

Lines \ell_{1} and \ell_{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 \ell, the transformation R(\ell) produces another line as follows: \ell is reflected in \ell_{1}, and the resulting line is then reflected in \ell_{2}. Let R^{(1)}(\ell)=R(\ell), and for integer n \geq 2 define R^{(n)}(\ell)= R\left(R^{(n-1)}(\ell)\right). Given that \ell is the line y=\frac{19}{92} x, find the smallest positive integer m for which R^{(m)}(\ell)=\ell.