HMMT November 2014 U18

For any positive integer $x$, define $\operatorname{Accident}(x)$ to be the set of ordered pairs $(s, t)$ with $s \in$ $\{0,2,4,5,7,9,11\}$ and $t \in\{1,3,6,8,10\}$ such that $x+s-t$ is divisible by 12 . For any nonnegative integer $i$, let $a_{i}$ denote the number of $x \in\{0,1, \ldots, 11\}$ for which $|\operatorname{Accident}(x)|=i$. Find

$$a_{0}^{2}+a_{1}^{2}+a_{2}^{2}+a_{3}^{2}+a_{4}^{2}+a_{5}^{2}$$