
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 \( \left(2 \sin x + a^2 + a \right)^3 - (\cos 2x + 3a \sin x + 11)^3 = 12 - 2 \sin^2 x + (3a - 2) \sin x - a^2 - a \) 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}
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