
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
Find all  $a\in\mathbb R$  for which there exists a function  $f\colon\mathbb R\rightarrow\mathbb R$ , such that
(i)  $f(f(x))=f(x)+x$ , for all  $x\in\mathbb R$ ,
(ii)  $f(f(x)-x)=f(x)+ax$ , for all  $x\in\mathbb R$ .

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