
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 circle $C: (x-a)^2 + (y-b)^2 = \frac{9}{4}$ whose center $C$ lies on the parabola $x^2 = 2py (p > 0)$, and the circle $C$ passes through the origin and is tangent to the directrix of the parabola.

$(1)$ Find the equation of this parabola;

$(2)$ A line $l$ passing through the focus $F$ of the parabola intersects the parabola at points $A$ and $B$. Tangents to the parabola at points $A$ and $B$ meet at point $P$. Find the minimum value of the area of triangle $PAB$ and the equation of line $l$ at this time.

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