
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 a parabola $y^{2}=4x$ with two fixed points $A$ and $B$ on either side of its axis of symmetry, and $F$ as the focus of the parabola. The distance between $F$ and $A$ is 2, and the distance between $F$ and $B$ is 5. Find a point $P$ on the parabolic segment $AOB$ such that the area of triangle $PAB$ is maximized. Also, find this maximum area.

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