Let circle $C_0$ be defined by the equation $x^2 + y^2 - 2xr - 2yr + r^2 = 0$ and circle $C_1$ by $x^2 + y^2 - 6xr - 2yr + 9r^2 = 0$. For all $i > 1$, circle $C_i$ is drawn such that it touches $C_0$, $C_{i - 1}$ and the $x$-axis.

Consider the case $r = 195643523275200$. Count the number of circles $C_i$ $(i \ge 0)$ having an integer radius.