
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
Let \( x \) and \( y \) be positive numbers and let \( s \) denote the smallest of the numbers \( x \), \( y + \frac{1}{x} \), and \( \frac{1}{y} \). What is the maximum possible value of \( s \), and for which values of \( x \) and \( y \) is it attained?

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