
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 moving circle $M$ that passes through the fixed point $F(0,-1)$ and is tangent to the line $y=1$. The trajectory of the circle's center $M$ forms a curve $C$. Let $P$ be a point on the line $l$: $x-y+2=0$. Draw two tangent lines $PA$ and $PB$ from point $P$ to the curve $C$, where $A$ and $B$ are the tangent points.
(I) Find the equation of the curve $C$;
(II) When point $P(x_{0},y_{0})$ is a fixed point on line $l$, find the equation of line $AB$;
(III) When point $P$ moves along line $l$, find the minimum value of $|AF|⋅|BF|$.

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