
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  $a$  be a given real number. Find all functions  $f : R \to  R$  such that  $(x+y)(f(x)-f(y))=a(x-y)f(x+y)$  holds for all  $x,y \in  R$ . 

(Walther Janous, Austria)

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