
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 $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, and $1+ \frac{\tan C}{\tan B}= \frac{2a}{b}$.
(1) Find the measure of angle $C$;
(2) If $\cos (B+ \frac{\pi}{6})= \frac{1}{3}$, find the value of $\sin A$;
(3) If $(a+b)^{2}-c^{2}=4$, find the minimum value of $3a+b$.

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