
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 the equation $x^{2}+4ax+3a+1=0$ ($a$ is a constant greater than $1$), its two roots are $\tan α$ and $\tan β$, where both $α$ and $β$ are within the interval $\left(-\frac{π}{2}, \frac{π}{2}\right)$. Find the value of $\tan \frac{α+β}{2}$.

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