
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 positive integer  $n$ , define  $S_n$  to be the minimum value of the sum \[ \sum_{k=1}^n \sqrt{(2k-1)^2+a_k^2}, \] where  $a_1,a_2,\ldots,a_n$  are positive real numbers whose sum is 17. There is 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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