
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 $C$: $\frac{y^{2}}{a^{2}} + \frac{x^{2}}{b^{2}} = 1 (a > b > 0)$ passing through the point $(-\frac{1}{2}, -\sqrt{3})$, with an eccentricity of $\frac{\sqrt{3}}{2}$.
(1) Find the standard equation of the ellipse $C$;
(2) A circle with the equation $x^{2} + y^{2} = 1$ has a tangent line $l$ passing through point $P(0, t)$. The line $l$ intersects the ellipse $C$ at points $A$ and $B$. Represent the area of $\triangle AOB$ (where $O$ is the coordinate origin) as a function $f(t)$, and find the maximum value of the function $f(t)$.

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