
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
For each positive integer  $n$  find the largest subset  $M(n)$  of real numbers possessing the property: \[n+\sum_{i=1}^{n}x_{i}^{n+1}\geq n \prod_{i=1}^{n}x_{i}+\sum_{i=1}^{n}x_{i}\quad \text{for all}\; x_{1},x_{2},\cdots,x_{n}\in M(n)\] When does the inequality become an equality ?

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