
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
Find all pairs of sets \(A\) and \(B\) such that \(A\) and \(B\) satisfy:
1. \(A \cup B = \mathbf{Z}\);
2. If \(x \in A\), then \(x-1 \in B\);
3. If \(x \in B\) and \(y \in B\), then \(x+y \in A\).

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