
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 in the polar coordinate system, the equation of curve $C_1$ is $\rho =2\sin (\theta +\frac{\pi }{3})$, and the equation of curve $C_2$ is $\rho \sin (\theta +\frac{\pi }{3})=4$. Establish a rectangular coordinate system $xOy$ with the polar point $O$ as the origin and the polar axis direction as the positive direction of the $x$-axis. Let $A,B$ be moving points on $C_1$ and $C_2$ respectively.

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