
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
A moving circle is externally tangent to the circle $C_{1}$: $x^{2}+y^{2}+6x+5=0$ and internally tangent to the circle $C_{2}$: $x^{2}+y^{2}-6x-91=0$. The locus of the center of the moving circle is curve $E$.  
$(1)$ Find the equation of curve $E$;  
$(2)$ Let point $P$ be a moving point on $E$, point $O$ be the origin, and the right focus of curve $E$ be $F$. Find the minimum value of $|PO|^{2}+|PF|^{2}$.

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