
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 sets $A = \{x \mid (x-2)(x-(3a+1)) < 0\}$ and $B = \{x \mid 2a < x < a^2 + 1\}$.  
(i) When $a = -2$, find $A \cup B$;  
(ii) Determine the range of real numbers $a$ such that $B \subseteq A$.

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