
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
Determine the values of \(a\) for which the curves \(x^2 + (y - 1)^2 = a^2\) and \(y = x^2 - a\) in the real \(xy\)-plane intersect at exactly 4 points.
A) $a > -\frac{1}{3}$ 
B) $a > 0$ 
C) $a = \frac{1}{3}$ 
D) $a = \frac{1}{2}$ 
E) $a \leq -3$

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