
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 a nonzero polynomial  $P(x,y)$  such that  $P(\lfloor a\rfloor,\lfloor 2a\rfloor)=0$  for all real numbers  $a.$ 

(Note:  $\lfloor v\rfloor$  is the greatest integer less than or equal to  $v.$ )

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