
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

Find all values of \(a\) for which the equation

\[
(\operatorname{tg} x + 6)^{2}-\left(a^{2}+2a+8\right)(\operatorname{tg} x + 6) + a^{2}(2a+8) = 0
\]

has exactly two solutions on the interval \(\left[0; \frac{3\pi}{2}\right]\).

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