
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 polar coordinate system, the equation of curve $C_1$ is $\rho=2\sin\left(\theta+ \frac{\pi}{3}\right)$, and the equation of curve $C_2$ is $\rho\sin\left(\theta+ \frac{\pi}{3}\right)=4$. Establish a Cartesian coordinate system $xOy$ with the pole $O$ as the origin and the direction of the polar axis as the positive direction of the x-axis.  
(1) Find the Cartesian equations of curves $C_1$ and $C_2$;  
(2) Let $A$ and $B$ be points on $C_1$ and $C_2$ respectively, find the minimum value of $|AB|$.

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