
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
Given \( x, y \in \mathbf{R} \) such that \( x^2 + y^2 = 2 \) and \( |x| \neq |y| \), find the minimum value of \( \frac{1}{(x+y)^2} + \frac{1}{(x-y)^2} \).

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