
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 $O$ as the pole and the positive $x$-axis as the polar axis, with matching units of length in both coordinate systems, the parametric equations for line $l$ are given by
$$
\begin{cases}
x = 1 + t\cos \alpha \\
y = t\sin \alpha
\end{cases}
\quad (t\text{ is a parameter}, 0 < \alpha < \pi),
$$
and the polar equation for curve $C$ is $\rho\sin^2\theta = 4\cos \theta$.

(I) Find the Cartesian (rectangular) coordinate equation of curve $C$.

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

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