
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
In the Cartesian coordinate system $xy$, a moving circle passes through the point $\left(\frac{1}{2}, 0\right)$ and is tangent to the line $x = -\frac{1}{2}$. Let the trajectory of the circle's center be curve $E$.
   (I) Find the equation for curve $E$.
   (II) Let $P$ be a moving point on curve $E$, and points $B$ and $C$ are on the y-axis. The incircle of $\triangle PBC$ has the equation $(x-1)^2+y^2=1$. Find the minimum area of $\triangle PBC$.

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