
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 the positive numbers \( x_{1}, x_{2}, \ldots, x_{n} \) is 1. Let

\[ s = \max \left( \frac{x_{1}}{1+x_{1}}, \frac{x_{2}}{1+x_{1}+x_{2}}, \ldots, \frac{x_{n}}{1+x_{1}+\cdots+x_{n}} \right) \]

What is the smallest possible value of \( s \)? For which values of \( x_{1}, x_{2}, \ldots, x_{n} \) does this occur?

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