
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 \((\cos 2x + 14 \cos x - 14a)^7 - (6a \cos x - 4a^2 - 1)^7 = (6a - 14)\cos x + 2 \sin^2 x - 4a^2 + 14a - 2\) has two different solutions on the interval \(\left[-\frac{2\pi}{3}; \pi\right]\). Indicate these solutions for each found \(a\).

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