
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 $E$ that passes through the point $F(1,0)$, and is tangent to the line $l: x=-1$.
(Ⅰ) Find the equation of the trajectory $G$ for the center $E$ of the moving circle.
(Ⅱ) Given the point $A(3,0)$, if a line with a slope of $1$ intersects with the line segment $OA$ (not passing through the origin $O$ and point $A$) and intersects the curve $G$ at points $B$ and $C$, find the maximum value of the area of $\triangle ABC$.

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