
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
The sum of \( n \) positive numbers \( x_{1}, x_{2}, \cdots, x_{n} \) is equal to 1. Let \( S \) be the largest number among the following: \( \frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}} \), \( \cdots, \frac{x_{n}}{1+x_{1}+x_{2}+\cdots+x_{n}} \). Find the minimum possible value of \( S \) and the values of \( x_{1}, x_{2}, \cdots, x_{n} \) for which this minimum value is achieved.

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