
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 the focus of the parabola $C_1$ coincides with the right focus of the ellipse $C_2: \frac{x^2}{6}+\frac{y^2}{5}=1$, and the vertex of the parabola $C_1$ is at the origin, a line $l$ passing through the point $M(4,0)$ intersects the parabola $C_1$ at points $A$ and $B$.

$(1)$ Write the standard equation of the parabola $C_1$;

$(2)$ Find the minimum value of the area of $\Delta ABO$.

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