
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 $P$ is a moving point on the hyperbola $\frac{x^2}{9} - \frac{y^2}{16} = 1$, point $M$ is a moving point on the circle $(x+5)^2 + y^2 = 4$, and point $N$ is a moving point on the circle $(x-5)^2 + y^2 = 1$, the maximum value of $|PM| - |PN|$ is \_\_\_\_\_\_.

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