
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 a function $f(x) = x^2$, let set $A = \{x | f(x - 3) < 1\}$, and set $B = \{x | f(x - 2a) < a^2\}$.
(1) Find the set $A$;
(2) Find the set $B$;
(3) If $A \cup B = B$, determine the range of the real number $a$.

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