
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 point $P(x,y)$ on the curve $C: \sqrt{\frac{x^2}{25}} + \sqrt{\frac{y^2}{9}} = 1$, and points ${F_1(-4,0)}$, ${F_2(4,0)}$, find the maximum value of $|PF_1| + |PF_2|$.

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