
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 $C$ with its vertex at the origin and its focus on the $x$-axis, and it passes through point $A(1,2)$.

$(I)$ Find the standard equation of $C$;

$(II)$ If $O$ is the origin, $F$ is the focus of $C$, and a line $l$ passing through $F$ with a slope of $45^\circ$ intersects $C$ at points $A$ and $B$, find the area of $\Delta AOB$.

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