
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, its upper vertex $M$ and left and right foci $F\_1$, $F\_2$ form a triangle $MF\_1F\_2$ with an area of $\sqrt{3}$. The eccentricity of the ellipse $C$ is $\frac{\sqrt{3}}{2}$.

1. Find the equation of the ellipse $C$.
2. The lower vertex of the ellipse $C$ is $N$, and the line passing through the point $T(t, 2)(t \neq 0)$ intersects the ellipse $C$ at points $E$ and $F$. If the area of triangle $TMN$ is $k$ times the area of triangle $TEF$, find the maximum value of $k$.

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