
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
Given a positive integer  $k$ , let  $||k||$  denote the absolute difference between  $k$  and the nearest perfect square. For example,  $||13||=3$   since the nearest perfect square to  $13$  is  $16$ . Compute the smallest positive integer  $n$  such that  $\frac{||1|| + ||2|| + ...+ ||n||}{n}=100$ .

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