
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 \( \sum_{k=1}^{n} \sqrt{(2 k-1)^{2} + a_{k}^{2}} \), where \( a_{1}, a_{2}, \cdots, a_{n} \in \mathbf{R}^{+} \) and \( \sum_{k=1}^{n} a_{k} = 17 \). If there exists \( n \) such that \( S_{n} \) is also an integer, find all possible values of \( n \).

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