$\frac{\pi}{70}$ and $\frac{\pi}{54}$ are the angles. For any line $l$ undergoing a transformation, denoted as $R$, resulting in another line $R(l)$, the transformation $R$ is: $l$ is first reflected over $l_{1}$, and the resulting line (i.e., the symmetric image of $l$ with respect to $l_{1}$) is then reflected over $l_{2}$, yielding $R(l)$. Let $R^{(1)}(l)=R(l)$, and for $n \geqslant 2$, define $R^{n}(l)=R\left(R^{n-1}(l)\right)$. Given: the line $l$ is $y=\frac{19}{92} x$. Find: the smallest positive integer $m$ such that $R^{(m)}(l)=l$.
Let $\lambda_{0}$ and $\lambda$ be two lines passing through the origin and making angles $\theta_{0}$ and $\theta$ with the $x$-axis, respectively. When $\lambda$ is reflected through $\lambda_{0}$, the resulting line $\lambda^{\prime}$ should make an angle $\theta_{0} + (\theta_{0} - \theta) = 2\theta_{0} - \theta$ with the $x$-axis. Thus, if $\lambda$ is reflected through $l_{1}$, the reflected line $\lambda_{1}$ is a line passing through the origin and making an angle $\frac{2\pi}{70} - \theta$ with the $x$-axis, and the line $\lambda_{1}$ reflected through $l_{2}$ results in a line $\lambda_{2}$ that passes through the origin and makes an angle $2 \cdot \frac{\pi}{54} - \left(\frac{2\pi}{70} - \theta\right) = \frac{8\pi}{945} + \theta$ with the $x$-axis. Therefore, $R(\lambda)$ should be the line obtained by rotating $\lambda$ by $\frac{8\pi}{945}$ radians about the origin, and $R^{(m)}(\lambda)$ is the line obtained by rotating $\lambda$ by $\frac{8m\pi}{945}$ radians. For $R^{(m)}(\lambda) = \lambda$ to hold, $\frac{8m}{945}$ must be an integer. Since $(945, 8) = 1$, the smallest value of $m$ should be 945.