
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 $C: y^2 = 2px (p > 0)$, and a circle $M: (x-2)^2 + y^2 = 4$, the distance from the center $M$ of the circle to the directrix of the parabola is $3$. Point $P(x_0, y_0)(x_0 \geqslant 5)$ is a point on the parabola in the first quadrant. Through point $P$, two tangent lines to circle $M$ are drawn, intersecting the x-axis at points $A$ and $B$.

$(1)$ Find the equation of the parabola $C$;
$(2)$ Find the minimum value of the area of $\triangle PAB$.

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