
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 point $F(0,1)$, moving point $M$ lies on the line $l:y=-1$. The line passing through point $M$ and perpendicular to the $x$-axis intersects the perpendicular bisector of segment $MF$ at point $P$. Let the locus of point $P$ be curve $C$.  
$(1)$ Find the equation of curve $C$;  
$(2)$ Given that the circle $x^{2}+(y+2)^{2}=4$ has a diameter $AB$, extending $AO$ and $BO$ intersect curve $C$ at points $S$ and $T$ respectively, find the minimum area of quadrilateral $ABST$.

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