
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
**4.** Find all functions of two variables defined over the entire plane that satisfy the relations  $f(x+u,y+u)=f(x,y)+u$  and  $f(xv,yv)= f(x,y) v$  for any real numbers  $x,y,u,v$ . **(R.12)**

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