
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 \( x \in \mathbf{R}, y \in \mathbf{R}^{+}, A=\left\{x^{2}+x+1,-x,-x-1\right\}, B=\left\{-y,-\frac{y}{2}, y+1\right\} \), if \( A = B \), determine the value of \( x^{2} + y^{2} \).

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