
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 a positive integer \( n \), define \( S_{n} \) as the minimum value of the sum
\[
\sum_{k=1}^{n} \sqrt{(2k-1)^{2}+a_{k}^{2}}
\]
where \( a_{1}, a_{2}, \cdots, a_{n} \) are positive real numbers whose sum is 17. There exists a unique positive integer \( n \) for which \( S_{n} \) is also an integer. Find this \( n \).

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