
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
Consider the ellipse (H): $\frac{x^2}{a^2} + y^2 = 1 (a > 1)$, the distance from the origin $O$ to the line $MN$ is $\frac{\sqrt{3}}{2}$, where the point $M(0, -1)$, and the point $N(a, 0)$.
(1) Find the eccentricity $e$ of the ellipse (H);
(2) The line $l$ passing through the right focus $F_2$ of the ellipse intersects the ellipse at points $A$ and $B$, and the point $C$ is on the ellipse and $O$ is the origin. If $\overrightarrow{OC} = \frac{1}{2}\overrightarrow{OA} + \frac{\sqrt{3}}{2}\overrightarrow{OB}$, find the equation of line $l$.

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