Let $\displaystyle{A_{0}BC_{0}D}$ be a convex quadrilateral inscribed in a circle $\displaystyle{\omega}$. For all integers $i\ge0$, let $P_{i}$ be the intersection of lines $A_{i}B$ and $C_{i}D$, let $Q_{i}$ be the intersection of lines $A_{i}D$ and $BC_{i}$, let $M_{i}$ be the midpoint of segment $P_{i}Q_{i}$, and let lines $M_{i}A_{i}$ and $M_{i}C_{i}$ intersect $\omega$ again at $A_{i+1}$ and $C_{i+1}$, respectively. The circumcircles of $\triangle{A_3M_3C_3}$ and $\triangle{A_4M_4C_4}$ intersect at two points $P$ and $M$. If $A_{0}B=3, BC_{0}=4, C_{0}D=6, DA_{0}=7,$ then $PM$ can be expressed in the form $\displaystyle{\frac{a\sqrt{b}}{c}}$ for positive integers $a, b, c$ such that $gcd(a, c)=1$ and $b$ is squarefree. Compute $100a+10b+c$. 

Note: a square-free integer (or squarefree integer) is an integer which is divisible by no perfect square other than 1. 

Source: Online Math Open