
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
Let's define a point on the plane $(u, v)$ as lying between the parabolas \( y = f(x) \) and \( y = g(x) \) if \( f(u) \leq v \leq g(u) \). Find the smallest real number \( p \) for which the following statement is true: any line segment whose endpoints and midpoint lie between the parabolas \( y = x^2 \) and \( y = x^2 + 1 \), lies entirely between the parabolas \( y = x^2 \) and \( y = x^2 + p \).

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