
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
Determine all nonzero integers  $a$  for which there exists two functions  $f,g:\mathbb Q\to\mathbb Q$  such that
\[f(x+g(y))=g(x)+f(y)+ay\text{ for all } x,y\in\mathbb Q.\]
Also, determine all pairs of functions with this property. 

*Vasile Pop*

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