
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 points $P(1, \frac{3\pi}{2})$ and $M(2, \frac{\pi}{6})$, line $l$ passes through point $P$ with a slope angle $\alpha = \frac{\pi}{3}$. Circle $M$ has center $M$ and radius $2$. Let $A$ and $B$ be the intersection points of line $l$ and circle $M$. Calculate the values of $|AB|$ and $|PA|\cdot|PB|$.

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