
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 maximum number  $ c$  such that for all  $n \in \mathbb{N}$  to have \[ \{n \cdot \sqrt{2}\} \geq \frac{c}{n}\] where  $ \{n \cdot \sqrt{2}\} \equal{} n \cdot \sqrt{2} \minus{} [n \cdot \sqrt{2}]$  and  $ [x]$  is the integer part of  $ x.$  Determine for this number  $ c,$  all  $ n \in \mathbb{N}$  for which  $ \{n \cdot \sqrt{2}\} \equal{} \frac{c}{n}.$ 

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