
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  $S$  be the set of all real numbers greater than or equal to  $1$ .
Determine all functions $ f: S \to S$ , so that for all real numbers  $x ,y  \in S$  with  $x^2 -y^2 \in S$  the condition  $f (x^2 -y^2) = f (xy)$  is fulfilled.

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