
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 a positive integer  $n$  be  $\textit{nice}$  if there exists a positive integer  $m$  such that \[ n^3 < 5mn < n^3 +100. \] Find the number of *nice* positive integers.

*Proposed by Akshaj*

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