
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
Given integer  $n\geq 2$  and positive real number  $a$ , find the smallest real number  $M=M(n,a)$ , such that for any positive real numbers  $x_1,x_2,\cdots,x_n$  with  $x_1 x_2\cdots x_n=1$ , the following inequality holds:
\[\sum_{i=1}^n \frac {1}{a+S-x_i}\leq M\]
where  $S=\sum_{i=1}^n x_i$ .

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