
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 an integer $n \geq 2$ and a positive real number $a$, let the positive real numbers $x_{1}, x_{2}, \ldots, x_{n}$ satisfy $\prod_{i=1}^{n} x_{i} = 1$. Find the minimum real number $M = M(n, a)$ such that $\sum_{i=1}^{n} \frac{1}{a+S-x_{i}} \leq M$ always holds, where $S = \sum_{j=1}^{n} x_{j}$.

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