
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
For which values of the parameter \(a\) does the equation \(\frac{\log_{a} x}{\log_{a} 2} + \frac{\log_{x} (2a - x)}{\log_{x} 2} = \frac{1}{\log_{a^2 - 1} 2}\) have (1) a solution? (2) exactly one solution?

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