11. Lines $l_{1}, l_{2}$ both pass through the origin, and in the first quadrant, they form angles of $\frac{\pi}{70}$ and $\frac{\pi}{54}$ with the positive $x$-axis, respectively. Define $R(l)$ as the line obtained by reflecting $l$ about $l_{1}$ and then reflecting the result about $l_{2}$. Let $R^{(1)}(l)=R(l), R^{\mathrm{a}}(l)$ $=R\left(R^{(n-1)}(l)\right)(n \geqslant 2)$. If the equation of line $l$ is $y=\frac{19}{92} x$, find the smallest positive integer $m_{0}$ such that $R^{(m)}(l)=l$.
A line passing through the origin and forming angles $\theta_{0}, \theta, \lambda$ with the positive x-axis. The symmetric line $\lambda^{\prime}$ about $\lambda$ forms an angle with the x-axis of $\theta_{0} + (\theta_{0} - \theta) = 2 \theta_{0} - \theta$.

Therefore, the symmetric line $\lambda_{1}$ about $l_{1}$ passes through the origin and forms an angle with the positive x-axis of $2 \cdot \frac{\pi}{70} - \theta$. The symmetric line $\lambda_{2}$ about $l_{2}$ passes through the origin and forms an angle with the positive x-axis of
$$
2 \cdot \frac{\pi}{54} - \left(2 \cdot \frac{\pi}{70} - \theta\right) = -\frac{8 \pi}{945} + \theta.
$$

Thus, $R(\lambda)$ can be obtained by rotating $\lambda$ by $-\frac{8 \pi}{945}$. Therefore, $R^{(m)}(\lambda)$ can be obtained by rotating $\lambda$ by $-\frac{8 m}{945}$.

If $\left.R^{(m)}\right)(\lambda) = \lambda$, then $\frac{8 m}{945}$ must be an integer. Therefore, the smallest positive integer $m$ is 945.