
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
\[\sum_{k=1}^n \sqrt{(3k-2)^2 + b a_k^2},\] where $a_1, a_2, \ldots, a_n$ are positive real numbers whose sum is $19$, and $b$ is a constant. Find the unique positive integer $n$ for which $T_n$ is also an integer, given $b=4$.

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