
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 a point O at the origin of the Cartesian coordinate system and the positive semi-axis of the x-axis as the polar axis, with the same length unit in both coordinate systems, the parametric equation of line l is given by $x = \frac{1}{2} + t\cos \theta$, $y = t\sin \theta$ ($t$ is a parameter, $0 < \theta < \pi$). The polar equation of curve C is $\rho\sin^2\alpha - 2\cos \alpha = 0$.

(1) Find the Cartesian equation of curve C.
(2) Let line l intersect curve C at points A and B. Find the minimum value of $|AB|$ as $\theta$ varies.

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