
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 $y^{2}=2px\left(p \gt 0\right)$, with its focus $F$ at a distance of $2$ from the directrix.<br/>$(1)$ Find the standard equation of the parabola;<br/>$(2)$ If $O$ is the origin, the slope is $2$, and the line $l$ passing through the focus $F$ intersects the parabola at points $A$ and $B$, find the area of $\triangle AOB$.

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