
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
If  $a,b,x,y$  are real numbers then find all solutions of  $a,b$  such that  $\left \lfloor ax+by \right \rfloor +  \left \lfloor bx+ay \right \rfloor = (a+b) \left \lfloor x+y \right \rfloor$ 

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