
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
A number \( n \) is called a "good number" if \( n \) is a perfect square or the distance from \( n \) to the nearest perfect square is a perfect cube. For example, \(\left|45^{2} - 2017\right| = 2^{3}\), so 2017 is a good number. For a positive integer \( N \), let \( S(N) \) be the number of good numbers in the set \(\{1, 2, \cdots, N\}\).

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

(2) Does there exist positive real numbers \( \alpha \) and \(\beta\) such that \( \lim_{N \rightarrow +\infty} \frac{S(N)}{N^{\alpha}} = \beta \)?

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