
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
In the Cartesian coordinate system, the eccentricity of the ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \ (a > b > 0)$ is $\frac{1}{2}$, and it passes through the point $(1,\frac{3}{2})$.

$(1)$ Find the equation of the ellipse $C$;

$(2)$ If a line $l$ passing through the point $T(0,1)$ intersects the ellipse at points $A$ and $B$, find the maximum area of $\triangle OAB$.

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