
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 a polar coordinate system, suppose a line passes through points A($\sqrt{3}$, $\frac{2\pi}{3}$) and B(3, $\frac{\pi}{2}$), and this line has exactly one common point with the curve C: $\rho = 2r\sin\theta$ ($r>0$). Find the value of the real number $r$.

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