
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 point $P$ lies on the hyperbola $\frac {x^{2}}{9}- \frac {y^{2}}{16}=1$. Points $M$ and $N$ are on the circles $(x+5)^2 + y^2 = 4$ and $(x-5)^2 + y^2 = 1$, respectively. Find the maximum value of $|PM| - |PN|$.

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