
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 vectors $a=\left( \sin\left( \alpha+\frac{\pi}{6} \right),3 \right)$ and $b=(1,4\cos \alpha)$, where $\alpha\in(0,\pi)$.

(1) If $a\perp b$, find the value of $\tan \alpha$;

(2) If $a\parallel b$, find the value of $\alpha$.

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