
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 the sets \(A=\left\{1, \frac{x+y}{2}-1\right\}\) and \(B=\{-\ln(xy), x\}\), if \(A = B\) and \(0 < y < 2\), find the value of 

\[ 
\left(x^{2} - \frac{1}{y^{2}}\right) + \left(x^{4} - \frac{1}{y^{4}}\right) + \cdots + \left(x^{2022} - \frac{1}{y^{2022}}\right).
\]

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