
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
For \( n \in \mathbf{N} \), let \( S_{n} \) be the minimum value of \( \sum_{k=1}^{n} \sqrt{(2k-1)^{2} + a_{k}^{2}} \), where \( a_{1}, a_{2}, a_{3}, \cdots, a_{n} \) are positive real numbers that sum to 17. If there exists a unique \( n \) such that \( S_{n} \) is an integer, find \( n \).

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