
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 that the two foci of ellipse C are $F_{1}(-\sqrt{3}, 0)$ and $F_{2}(\sqrt{3}, 0)$, and ellipse C passes through point $P(1, \frac{\sqrt{3}}{2})$.
(1) Find the standard equation of ellipse C.
(2) If a line parallel to OP intersects ellipse C at points A and B, and OA is perpendicular to OB, find the area of triangle AOB.

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