
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
Let $P$ be a point on the right branch of the hyperbola $x^{2} - \frac{y^{2}}{15} = 1$. Let $M$ and $N$ be points on the circles 
$$
(x + 4)^{2} + y^{2} = 4 \quad \text{and} \quad (x - 4)^{2} + y^{2} = 1
$$
respectively. Find the maximum value of $|PM| - |PN|$.

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