
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

The numbers \(1, 2, 3, \ldots, n\) are written in some order: \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\). Consider the sum \(S = \frac{a_{1}}{1} + \frac{a_{2}}{2} + \ldots + \frac{a_{n}}{n}\). Find such an \(n\) so that among these sums (with all possible permutations of \(a_{1}, a_{2}, a_{3}, \ldots, a_{n}\)) all the integers from \(n\) to \(n+100\) occur.

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