
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 fixed point $F(0,1)$ and a fixed line $l: y=-1$, a moving circle $M$ passes through point $F$ and is tangent to line $l$.  
(Ⅰ) Find the equation of the trajectory $C$ of the center of circle $M$;  
(Ⅱ) A line passing through point $F$ intersects curve $C$ at points $A$ and $B$. Tangents to curve $C$ at points $A$ and $B$ are drawn, intersecting at point $P$. Find the minimum area of the circumcircle of $\triangle PAB$.

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