
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 $x^2+4ax+3a+1=0$ (where $a$ is a constant greater than 1) has roots $\tan\alpha$ and $\tan\beta$, and $\alpha, \beta \in \left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$, then the value of $\tan \frac{\alpha +\beta}{2}$ is __________.

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