Let $a$ be a real number, and let $C$ be the circumference of the circle with center $(0,\, a)$ and radius $1$ in the coordinate plane.
[(1)] Find the range of $a$ such that $C$ is entirely contained in the region represented by the inequality $y > x^2$.
[(2)] Suppose $a$ is in the range found in (1). Let $S$ be the part of $C$ satisfying $x \geq 0$ and $y < a$. For a point $\mathrm{P}$ on $S$, let $L_{\mathrm{P}}$ be the length of the chord cut off from the tangent line to $C$ at $\mathrm{P}$ by the parabola $y = x^2$. Find the range of $a$ such that there exist two distinct points $\mathrm{Q}$, $\mathrm{R}$ on $S$ satisfying $L_{\mathrm{Q}} = L_{\mathrm{R}}$.
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Let $a$ be a real number, and let $C$ be the circumference of the circle with center $(0,\, a)$ and radius $1$ in the coordinate plane.
\begin{enumerate}
\item[(1)] Find the range of $a$ such that $C$ is entirely contained in the region represented by the inequality $y > x^2$.
\item[(2)] Suppose $a$ is in the range found in (1). Let $S$ be the part of $C$ satisfying $x \geq 0$ and $y < a$. For a point $\mathrm{P}$ on $S$, let $L_{\mathrm{P}}$ be the length of the chord cut off from the tangent line to $C$ at $\mathrm{P}$ by the parabola $y = x^2$. Find the range of $a$ such that there exist two distinct points $\mathrm{Q}$, $\mathrm{R}$ on $S$ satisfying $L_{\mathrm{Q}} = L_{\mathrm{R}}$.
\end{enumerate}
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