
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 $P$ moves on the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$, and points $A$ and $B$ move respectively on the circles $x^2 + (y-4)^2 = 16$ and $x^2 + (y+4)^2 = 4$. The maximum value of $PA + PB$ is \_\_\_\_\_\_.

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