
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
If the focus of the parabola $y^2=2px$ ($p>0$) is $F$, and point $A(3, 2)$ is inside the opening of the parabola, and point $P$ is a point on the parabola, given that the perimeter of $\triangle APF$ is minimized when the area of $\triangle APF$ is 1, calculate the length of $|PF|$.

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