
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 \( M = \left\{(x, y) \mid y = x^2 \right\}, N = \left\{(x, y) \mid x^2 + (y - a)^2 = 1 \right\} \), and \( A = M \cap N \):
(1) If \( |A| = 3 \), find the value of the real number \( a \).
(2) If \( A = \varnothing \), determine the range of values for the real number \( a \).

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