
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  $c > 1$  be a real number. A function  $f: [0 ,1 ] \to R$  is called c-friendly if  $f(0) = 0, f(1) = 1$  and  $|f(x) -f(y)| \le c|x - y|$  for all the numbers  $x ,y \in [0,1]$ . Find the maximum of the expression  $|f(x) - f(y)|$  for all *c-friendly* functions   $f$  and for all the numbers  $x,y \in [0,1]$ . 

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