
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
In the Cartesian coordinate system $(xOy)$, the parametric equations of line $l$ are given by: $\begin{cases}x=a-\frac{1}{2}t \\ y=\frac{\sqrt{3}}{2}t\end{cases} (t \text{ is a parameter})$, and in the polar coordinate system with pole $O$ and polar axis as the non-negative half of the $x$-axis, the polar equation of curve $C$ is given by $\rho \cos^2 \theta = \sin \theta$. The line $l$ intersects the curve $C$ at points $M$ and $N$ ($M$ is above $N$).
(I) If $a=0$, find the polar coordinates of points $M$ and $N$.
(II) If $P(a,0)$, and $|PM| + |PN| = 8 + 2\sqrt{3}$, find the value of $a$.

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