
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 the sets \( A, B \subset \mathbb{R} \). For any \( \alpha > 0 \), it holds that \( A \subseteq B + \alpha \mathbb{Z} \) and \( B \subseteq A + \alpha \mathbb{Z} \). 

(1) Does \( A = B \) necessarily hold?

(2) If \( B \) is a bounded set, does (1) hold? 

Note: \( X + \alpha \mathbb{Z} = \{ x + \alpha n \mid x \in X, n \in \mathbb{Z} \} \).

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