
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Given a circle $C$ with its center on the line $f(x) \leqslant g(x)$, and $F(x) = a^{f(x)} + tx^2 - 2t + 1$ axis, the lengths of the chords intercepted by circle $C$ on the $x$-axis and $y$-axis are $2\sqrt{5}$ and $4\sqrt{2}$, respectively. If the center $C$ is located in the fourth quadrant.

$(1)$ Find the equation of circle $C$;

$(2)$ Let the midpoint of chord $AB$ intercepted by circle $C$ on the $F(x) = a^{f(x)} + tx^2 - 2t + 1$ axis be $N$, and point $P$ inside circle $C$ makes $PA$, $PN$, $PB$ form a geometric sequence. Find the range of values for $\overrightarrow{PA} \cdot \overrightarrow{PB}$.

Present the answer in LaTex format: \boxed{Your answer}
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