
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 range of values for $b$ such that the circle $x^2 + y^2 = b^2$ intersects the parabola given by $y = x^2 - 2b$ at exactly three points in the real $xy$-plane.
A) $b \leq \frac{1}{4}$
B) $\frac{1}{4} < b \leq \frac{1}{3}$
C) $b > \frac{1}{4}$
D) $b = \frac{1}{2}$
E) $b > \frac{1}{2}$

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