
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 point $(1,0)$ and is tangent to the line $x=-1$, let the locus of the center of circle $M$ be curve $E$.

(I) Find the equation of curve $E$.

(II) If point $A$ has coordinates $(4,0)$, and points $B,C$ are two points on curve $E$ such that $A,B,C$ form an equilateral triangle, find the length of $\Delta ABC$.

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