
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 $F$ is the right focus of the ellipse $C:\frac{x^2}{4}+\frac{y^2}{3}=1$, $P$ is a point on the ellipse $C$, and $A(1,2\sqrt{2})$, find the maximum value of $|PA|+|PF|$.

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