
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 natural numbers \(a, b,\) and \(c\) such that for any \(n \in \mathbf{N}$, \(n > 2\), the inequality holds:
$$
b - \frac{c}{(n-2)!} < \frac{2^{3} - a}{2!} + \frac{3^{3} - a}{3!} + \cdots + \frac{n^{3} - a}{n!} < b.
$$

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