
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 a selected topic of *Coordinate System and Parametric Equations* from the advanced mathematics textbook 4-4, the origin O is taken as the pole and the positive half of the x-axis as the initial line (polar axis). Both the polar and the rectangular (Cartesian) coordinate systems share the same unit length. Given that the parameter equation of the line l is

$$
\begin{cases}
x = \frac{1}{2} + t \cos \alpha \\
y = t \sin \alpha
\end{cases}
$$

(where t is the parameter, and $0 < \alpha < \pi$), and the polar equation of the curve C is

$$
\rho = \frac{2 \cos \theta}{\sin^2 \theta},
$$

(I) Find the rectangular coordinate equation for curve C.

(II) Assuming that line l intersects curve C at points A and B, find the minimum value of |AB| as $\alpha$ varies.

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