
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 the sets \( A = \{(x, y) \mid a x + y = 1\} \), \( B = \{(x, y) \mid x + a y = 1\} \), and \( C = \{(x, y) \mid x^2 + y^2 = 1\} \), determine:
(1) For what values of \( a \) does \( (A \cup B) \cap C \) contain exactly two elements?
(2) For what values of \( a \) does \( (A \cup B) \cap C \) contain exactly three elements?

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