
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 integers \( n \), 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 20. Find the unique positive integer \( n \) for which \( S_n \) is also an integer.

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