
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 coordinate system with the origin $O$ as the pole and the positive half of the $x$-axis as the polar axis, and both coordinate systems have equal unit lengths. The parametric equations of line $l$ are given by $\begin{cases} x = t \cos{\phi} \\ y = 2 + t \sin{\phi} \end{cases}$ where $t$ is the parameter and $0 \leqslant \phi < \pi$. The polar coordinate equation of curve $C$ is given by $\rho \cos^2{\theta} = 8 \sin{\theta}$.

1. Find the Cartesian equation of line $l$ and curve $C$.
2. Suppose line $l$ intersects with curve $C$ at points $A$ and $B$. Find the minimum value of $|AB|$ as $\phi$ varies.

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