
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 point $M$ whose distance to the point $F(0,1)$ is equal to its distance to the line $y=-1$, the trajectory of point $M$ is denoted as $C$.

(1) Find the equation of the trajectory $C$.

(2) Let $P$ be a point on the line $l: x-y-2=0$. Construct two tangents $PA$ and $PB$ from point $P$ to the curve $C$.

  (i) When the coordinates of point $P$ are $\left(\frac{1}{2},-\frac{3}{2}\right)$, find the equation of line $AB$.

  (ii) When point $P(x_{0},y_{0})$ moves along the line $l$, find the minimum value of $|AF|\cdot|BF|$.

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