
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 $C: y^2 = 2px$ ($p > 0$) and a point $M(3, y_0)$ on $C$ such that the distance from $M$ to the focus $F$ is equal to 4.
(Ⅰ) Find the equation of the parabola $C$;
(Ⅱ) If a line $l$ passing through the point $(4, 0)$ intersects the parabola $C$ at points $A$ and $B$, find the minimum value of the area of $\triangle ABO$.

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