
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
Consider the parabola $y = Bx^2$ and the curve given by $y^2 - 2y = x^2 + 2y + 1$. Determine how many points do these two graphs intersect when $B$ is a positive constant, and $x$ and $y$ are real variables.
A) Exactly 4
B) Exactly 2
C) At least 1, but varies with $B$
D) 0 for at least one value of $B$
E) None of these

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