
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 equation of circle $C$: $x^{2}+y^{2}=4$, two tangent lines are drawn from point $M(2,4)$ to the circle, with tangent points $A$ and $B$. Line segment $AB$ passes through the right vertex and upper vertex of ellipse $T$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$.

1. Find the equation of ellipse $T$.
2. Given that line $l$: $y=kx+\sqrt{3} (k > 0)$ intersects the ellipse at points $P$ and $Q$, and $O$ is the coordinate origin, find the maximum area of $\triangle OPQ$.

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