
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
For positive numbers \( x \) and \( y \), \( a = \frac{1}{y} \) and \( b = y + \frac{1}{x} \) are calculated. Then, \( C \) is found to be the smallest number among the three: \( x \), \( a \), and \( b \).

What is the maximum value that \( C \) can take?

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