
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 slope angle is $\alpha$ and line $l$ passes through point $M(\sqrt{3}, 0)$, it intersects with the ellipse C: $\frac{x^2}{4} + y^2 = 1$ at points A and B.
(1) If $\alpha = \frac{\pi}{3}$, write the parametric equations of line $l$ and ellipse C.
(2) If $\frac{|OM|}{|AB|} = \frac{\sqrt{3}}{3}$, find the equation of line $l$.

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