
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
In the coordinate plane $xOy$, let $AB$ be a chord of the parabola $y^2 = 4x$ passing through the point $F(1,0)$. The circumcircle of the triangle $\triangle AOB$ intersects the parabola at point $P$ (distinct from points $O$, $A$, and $B$). If $PF$ bisects the angle $\angle APB$, find all possible values of $|PF|$.

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