
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
Let $x$ and $y$ be complex numbers such that
\[\frac{x + ky}{x - ky} + \frac{x - ky}{x + ky} = 1.\]
Find
\[\frac{x^6 + y^6}{x^6 - y^6} + \frac{x^6 - y^6}{x^6 + y^6}.\]
where $k$ is a real number.

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