
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
Find all numbers \( d \in (0, 1] \) that have the following property: if \( f(x) \) is an arbitrary continuous function defined for \( x \in [0, 1] \), where \( f(0) = f(1) \), then there exists a number \( x_{0} \in [0, 1-d] \) such that

\[ f\left(x_{0}\right) = f\left(x_{0} + d\right) \]

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