
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 $T_n$ to be the minimum value of the sum
\[\sum_{k=1}^n \sqrt{(3k-2)^2+b_k^2},\]where $b_1,b_2,\ldots,b_n$ are positive real numbers whose sum is $23$. Find the unique positive integer $n$ for which $T_n$ is also an integer.

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