
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 \( f: \mathbb{R} \rightarrow \mathbb{R} \) be a differentiable function such that \( f(0) = 0 \), \( f(1) = 1 \), and \( \left| f^{\prime}(x) \right| \leq 2 \) for all real numbers \( x \). If \( a \) and \( b \) are real numbers such that the set of possible values of \( \int_{0}^{1} f(x) \, dx \) is the open interval \( (a, b) \), determine \( b - a \).

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