
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 all positive real numbers  $c$  such that there are infinitely many pairs of positive integers  $(n,m)$  satisfying the following conditions:  $n \ge m+c\sqrt{m - 1}+1$  and among numbers  $n. n+1,.... 2n-m$  there is no square of an integer.

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