
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 set A = {x | (x - 2)(x - 3a - 2) < 0} and set B = {x | (x - 1)(x - a² - 2) < 0}, where a > 0:
1. Find A ∩ B when a = 1.
2. Let p be the proposition "x ∈ A" and q be the proposition "x ∈ B". If q is a necessary condition for p, find the range of real number values for a.

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