
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
Today, Ivan the Confessor prefers continuous functions  $f:[0,1]\to\mathbb{R}$  satisfying  $f(x)+f(y)\geq |x-y|$  for all pairs  $x,y\in [0,1]$ . Find the minimum of  $\int_0^1 f$  over all preferred functions.

(Proposed by Fedor Petrov, St. Petersburg State University)

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