
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 parametric equations of curve $C_1$: $$\begin{cases} x=\cos\theta \\ y=1+\sin\theta \end{cases}$$ (where $\theta$ is the parameter), and establishing a polar coordinate system with the origin as the pole and the positive half-axis of the x-axis as the polar axis, the polar equation of curve $C_2$ is: $\rho=4\sin\left(\theta+ \frac{\pi}{3}\right)$, and the polar equation of line $l$ is $\theta= \frac{\pi}{6}$.  
(1) Find the standard equation of curve $C_1$ and the Cartesian coordinate equation of curve $C_2$;  
(2) If curve $C_1$ and curve $C_2$ intersect line $l$ at points A and B, respectively, which are not the origin, find the value of $|AB|$.

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