
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
Find the smallest real number  $C$ , such that for any positive integers  $x \neq y$  holds the following:  $$ \min(\{\sqrt{x^2 + 2y}\}, \{\sqrt{y^2 + 2x}\})<C $$  Here  $\{x\}$  denotes the fractional part of  $x$ . For example,  $\{3.14\} = 0.14$ .

*Proposed by Anton Trygub*

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