
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
In the Cartesian coordinate system $xoy$, the parametric equation of circle $C$ is 
$$
\begin{cases} 
x = -1 + 4\cos\theta \\
y = 2 + 4\sin\theta 
\end{cases}
$$
($\theta$ is the parameter). Taking the origin $O$ as the pole and the non-negative half-axis of $x$ as the polar axis, the polar equation of line $l$ is $\sqrt{2}\rho\sin(\theta + \frac{3\pi}{4}) = 7$.

(1) Find the Cartesian coordinate equation of line $l$;  
(2) $A$ and $B$ are moving points on circle $C$ and line $l$, respectively. Find the minimum value of $|AB|$.

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