
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
A parabola with equation $y^2 = px$ ($p > 0$) has a focus at point $F$, and a slanted line passing through focus $F$ intersects the parabola at points $A$ and $B$. The area of $\triangle OAB$ is $2\sqrt{2}$, where $O$ is the origin. Find the value of $p$ in the equation of the parabola.

A) $y^2 = 2x$
B) $y^2 = 2\sqrt{2}x$
C) $y^2 = 4x$
D) $y^2 = 4\sqrt{2}x$

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