
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 vertex of the parabola $C$ is at the origin, the focus is on the positive $y$-axis, and the line $l: mx+y-\frac{3}{2}=0$ passes through the focus of the parabola $C$.  
$(1)$ Find the equation of the parabola $C$;  
$(2)$ If the line $l$ intersects the parabola $C$ at points $A$ and $B$, and tangents to the parabola $C$ are drawn through points $A$ and $B$, with the two tangents intersecting at point $P$, find the minimum area of $\triangle ABP$.

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