Let $f(x) = -\dfrac{\sqrt{2}}{4}x^2 + 4\sqrt{2}$. For a real number $t$ satisfying $0 < t < 4$, let $C_t$ be the circle that passes through the point $(t, f(t))$ in the coordinate plane, has a common tangent line with the parabola $y = f(x)$ at this point, and has its center on the $x$-axis.
[(1)] Let the center of circle $C_t$ have coordinates $(c(t), 0)$ and radius $r(t)$. Express $c(t)$ and $\{r(t)\}^2$ as polynomials in $t$.
[(2)] Suppose the real number $a$ satisfies $0 < a < f(3)$. How many real numbers $t$ in the range $0 < t < 4$ are there such that the circle $C_t$ passes through the point $(3, a)$?
Let $f(x) = -\dfrac{\sqrt{2}}{4}x^2 + 4\sqrt{2}$. For a real number $t$ satisfying $0 < t < 4$, let $C_t$ be the circle that passes through the point $(t, f(t))$ in the coordinate plane, has a common tangent line with the parabola $y = f(x)$ at this point, and has its center on the $x$-axis.
\begin{enumerate}
\item[(1)] Let the center of circle $C_t$ have coordinates $(c(t), 0)$ and radius $r(t)$. Express $c(t)$ and $\{r(t)\}^2$ as polynomials in $t$.
\item[(2)] Suppose the real number $a$ satisfies $0 < a < f(3)$. How many real numbers $t$ in the range $0 < t < 4$ are there such that the circle $C_t$ passes through the point $(3, a)$?
\end{enumerate}