
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
Point \(P\) lies on the right branch of the hyperbola \(x^{2}- \frac{y^{2}}{15}=1\). Points \(M\) and \(N\) lie respectively on the circles \((x+4)^{2}+y^{2}=4\) and \((x-4)^{2}+y^{2}=1\). Find the maximum value of \(|PM|-|PN|\).

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