
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 the function \( f(x) \) be defined on \([0,1]\), satisfying \( f(0) = f(1) \) and for any \( x, y \in [0,1] \) having \( |f(x) - f(y)| < |x - y| \). Find the smallest real number \( m \) such that for any function \( f(x) \) meeting the above conditions and for any \( x, y \in [0,1] \), we have \( |f(x) - f(y)| < m \).

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