
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: $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$. A sum $S = \frac{a_{1}}{1} + \frac{a_{2}}{2} + \ldots + \frac{a_{n}}{n}$ is taken. Find such $n$ so that among these sums (for all possible permutations of $a_{1}, a_{2}, a_{3}, \ldots, a_{n}$), all integer numbers from $n$ to $n + 100$ are encountered.

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