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

(1) Find the Cartesian equation of curve \(C\).

(2) Assume that line \(l\) intersects curve \(C\) at points \(A\) and \(B\). As \(θ\) varies, find the minimum value of \(|AB|\).

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