
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 triangle $ABC$, the sides opposite to angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it is given that $\frac{\cos A}{a} + \frac{\cos B}{b} = \frac{2\sqrt{3}\sin C}{3a}$.
$(1)$ Find the magnitude of angle $B$;
$(2)$ If the area of triangle $ABC$ is $\frac{\sqrt{3}}{2}$ and $B$ is an obtuse angle, find the minimum value of $b$.

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