
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
Determine all positive integers \( N \) for which there exists a strictly increasing sequence of positive integers \( s_{0} < s_{1} < s_{2} < \cdots \) satisfying the following properties:
- The sequence \( s_{1} - s_{0}, s_{2} - s_{1}, s_{3} - s_{2}, \ldots \) is periodic; and
- \( s_{s_{n}} - s_{s_{n-1}} \leq N < s_{1+s_{n}} - s_{s_{n-1}} \) for all positive integers \( n \).

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