
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
A circle is inscribed within the parabola described by the equation \( y = 2x^2 + 4x \) so that it is tangent to the parabola at two points. How much higher is the center of the circle than the points of tangency?

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