
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
In the Cartesian coordinate system $xOy$, the vertex of parabola $C$ is at the origin, and the coordinates of the focus $F$ are $(1,0)$.

$(1)$ Find the standard equation of parabola $C$;

$(2)$ A line $l$ passing through the focus $F$ of the parabola intersects the parabola at points $A$ and $B$. Find the minimum value of the area of $\Delta ABO$.

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