The Unit Circle and Circular Functions · Superchallenge problem

For integers $a$ and $b$ with $0 \le a \le 11$ and $0 \le b \le 11$, let $M$ be the midpoint of the segment joining the terminal points $P\!\left(\dfrac{a\pi}{6}\right)$ and $P\!\left(\dfrac{b\pi}{6}\right)$ (if $a = b$, $M$ is that point itself). For how many of the $144$ ordered pairs $(a, b)$ is $M$ less than $\dfrac12$ unit from the origin?