Let $S$ be the set of lattice points in the coordinate plane both of whose coordinates are integers between 1 and 20, inclusive. Exactly 133 points in $S$ lie on or below a line with equation $y=mx$. The possible values of $m$ lie in an interval of length $\frac{a}{b}$, where $a$ and $b$ are relatively prime positive integers. What is $a+b$?