
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 the parabola $y^2 = 8(x+2)$, find the length of the segment $PF$, where $F$ is the focus of the parabola and $P$ is the point on the $x$-axis that is the intersection of the perpendicular bisector of segment $AB$, and $A$ and $B$ are the points of intersection of the parabola and the line with an inclination angle of $60^{\circ}$ through the focus $F$.

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