
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 \(a\) do the roots \(x_{1}, x_{2}\) of the equation \(x^{2} - 2a x - \frac{1}{a^{2}} = 0\) satisfy the equality \(x_{1}^{4} + x_{2}^{4} = 16 + 8\sqrt{2}\)?

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