
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
Choose one of the following two questions to answer. If both are answered, only the score for question (A) will be counted.

A. (Inequality Choice Question) There are three propositions: ① If $|a-b| < 1$, then $|a| < |b|+1$; ② If $a, b \in \mathbb{R}$, then $|a+b|-2|a| \leqslant |a-b|$; ③ If $|x| < 2$, $|y| > 3$, then $\left| \frac{x}{y}\right| < \frac{2}{3} $; Among these, the correct ones are _______ (Fill in the serial numbers)

B. (Coordinate System and Parametric Equation Choice Question) In the polar coordinate system, the polar equation of line $l$ is $\rho \sin (\theta -\frac{\pi }{6})=3$, the minimum distance from point $A$ with polar coordinates $(2, \frac{\pi}{3})$ to a point on line $l$ is _______.

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