
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 the parameter \( a \) for which the equation \(\left(2 \sin x + a^{2} + a\right)^{3} - (\cos 2 x + 3 a \sin x + 11)^{3} = 12 - 2 \sin ^{2} x + (3 a - 2) \sin x - a^{2} - a\) has two distinct solutions on the interval \(\left[-\frac{\pi}{6} ; \frac{3 \pi}{2}\right]\). Indicate these solutions for each found \( a \).

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