
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 real numbers \(a\) and \(b\) so that for all real numbers \(x\),
\[ 2 \cos^{2}\left(x+\frac{b}{2}\right) - 2 \sin\left(a x - \frac{\pi}{2}\right) \cos\left(a x - \frac{\pi}{2}\right) = 1 \]

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