
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 $\sin \beta + 2\sin (2\alpha + \beta) = 0$, and $\alpha \neq \frac{k\pi}{2}, \alpha + \beta \neq \frac{\pi}{2} + k\pi \ (k \in \mathbb{Z})$, find the value of $3\tan (\alpha + \beta) + \tan \alpha$.

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