
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
Consider a polynomial  $P(x,y,z)$  in three variables with integer coefficients such that for any real numbers  $a,b,c,$    $$ P(a,b,c)=0 \Leftrightarrow a=b=c. $$  Find the largest integer  $r$  such that for all such polynomials  $P(x,y,z)$  and integers  $m,n,$    $$ m^r\mid P(n,n+m,n+2m). $$  [i]Proposed by Ma Zhao Yu

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