
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 $(C)$ that passes through point $F(0,1)$, with its center $(C)$ located above the $x$-axis. The distance from $(C)$ to point $F$ is greater than its distance to the $x$-axis by $1$.
(I) Find the equation of the trajectory $(E)$ of the moving circle's center.
(II) Suppose $A$ and $B$ are two distinct points on curve $(E)$. The tangents at $A$ and $B$ intersect at $P$, and $AP ⊥ BP$. Find the minimum value of $|AB|$.

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