
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

For what values of the parameter \( a \in \mathbb{R} \) does the maximum distance between the roots of the equation

\[ a \tan^{3} x + \left(2-a-a^{2}\right) \tan^{2} x + \left(a^{2}-2a-2\right) \tan x + 2a = 0 \]

belonging to the interval \(\left(-\frac{\pi}{2} , \frac{\pi}{2}\right)\), take the smallest value? Find this smallest value.

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