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

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