
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 that the moving circle $P$ is tangent to the circle $M: (x+3)^{2}+y^{2}=1$ externally and tangent to the circle $N: (x-3)^{2}+y^{2}=81$ internally.  
$(1)$ Find the equation of the trajectory of the center of the moving circle $P$.  
$(2)$ Find the range of values of $\frac{1}{PM}+\frac{1}{PN}$.

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