
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 $A(0,3\sqrt{7})$, the left focus of the hyperbola $E:\frac{x^2}{2}-\frac{y^2}{7}=1$ is $F$, and point $P$ moves on the right branch of the hyperbola $E$. Find the value of $|AP|+|PF|$ when the perimeter of $\triangle APF$ is minimized.

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