
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 an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, with its left and right foci being $F_1$ and $F_2$ respectively, and its eccentricity being $\frac{1}{2}$. Point $A$ is on the ellipse $C$, with $|AF_1| = 2$, and $\angle F_1AF_2 = 60^\circ$. A line $l$ that is not perpendicular to the coordinate axes passes through $F_2$ and intersects the ellipse $C$ at points $P$ and $Q$, with $N$ being the midpoint of $P$ and $Q$.
- (I) Find the equation of the ellipse $C$;
- (II) Given point $M(0, \frac{1}{8})$, and $MN \perp PQ$, find the equation of the line where $MN$ lies.

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