
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 polar coordinate system with the origin O as the pole and the positive x-axis as the polar axis, using the same unit length for both coordinate systems, the parametric equation of line l is given by:
$$
\left\{\begin{matrix} x= \frac{1}{2} + t\cos\theta \\ y= t\sin\theta \end{matrix}\right\},
$$
where $t$ is the parameter and $0 < \theta < \pi$. The polar equation of curve C is
$$
\rho\sin^2\theta - 2\cos\theta = 0.
$$

1. Find the rectangular coordinate equation of curve C.
2. Suppose the line l intersects the curve C at points A and B. As $\theta$ varies, find the minimum value of $|AB|$.

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