
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 all integer  $n$  such that the following property holds: for any positive real numbers  $a,b,c,x,y,z$ , with  $max(a,b,c,x,y,z)=a$  ,  $a+b+c=x+y+z$  and  $abc=xyz$ , the inequality   $$ a^n+b^n+c^n \ge x^n+y^n+z^n $$   holds.

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