
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 the ellipse $E: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{3}=1(a > \sqrt{3})$ with an eccentricity of $e= \frac{1}{2}$. The line $x=t(t > 0)$ intersects the curve $E$ at two distinct points $M$ and $N$. A circle $C$ is drawn with the segment $MN$ as its diameter, and its center is at $C$.

1. Find the equation of the ellipse $E$.
2. If the circle $C$ intersects the $y$-axis at two distinct points $A$ and $B$, find the maximum area of $\triangle ABC$.

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