
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 \( d \in (0,1] \) with the following property: If \( f(x) \) is any continuous function defined on \([0,1]\) such that \( f(0) = f(1) \), then there exists \( x_0 \in [0, 1-d] \) such that
\[ f(x_0) = f(x_0 + d). \]

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