
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:
[list=disc]
[*]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  $n$ [/list]

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