
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 an ellipse $(E)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with a minor axis of $2$ and an eccentricity of $\frac{\sqrt{6}}{3}$, the line $x=my-1 (m \in \mathbb{R})$ intersects the ellipse $(E)$ at points $A$ and $B$, with $O$ as the coordinate origin.

1. Find the equation of the ellipse $(E)$.
2. Find the maximum area of $\triangle OAB$.
3. Determine the position relationship between the point $G(-2, 0)$ and the circle with $AB$ as its diameter when $m \in \mathbb{R}$, and explain the reasoning.

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