
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
It is known that F is the intersection point of the parabola $C: y^2 = 2px$ ($p > 0$) and the line $l_1: y = -x$, with one intersection point of the parabola $C$ and the line $l_1$ having an x-coordinate of 8.
(1) Find the equation of the parabola $C$;
(2) A line $l_2$, which does not pass through the origin and is perpendicular to $l_1$, intersects the parabola at two distinct points $A$ and $B$. If the midpoint of the line segment $AB$ is $P$, and $|OP| = \frac{1}{2}|AB|$, find the area of $\triangle FAB$.

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