
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 that $O$ is the origin, the equation of the parabola $C$ is $y^{2}=8x$. $F$ is the focus of $C$, $A(2,4)$, and the line $l$ passing through point $F$ intersects the parabola $C$ at points $P$ and $Q$ (other than point $A$). $AP$ and $AQ$ intersect the $x$-axis at points $M$ and $N$, respectively. Find $|OM|\cdot |ON|$.

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