
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
For a positive integer \( n \) that is either a perfect square or the distance from \( n \) to the nearest perfect square is a perfect cube, \( n \) is defined as a "good number." For example, \(|45^2 - 2017| = 2^3\), which makes 2017 a good number. For a given positive integer \( N \), let \( S(N) \) be the number of good numbers in the set \(\{1, 2, \cdots, N\}\).

(1) Find \( S(2018) \).

(2) Determine whether there exist positive real numbers \(\alpha\) and \(\beta\) such that
\[
\lim_{{N \to \infty}} \frac{S(N)}{N^{\alpha}} = \beta.
\]

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