
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
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and the point $\left( n,\frac{S_n}{n} \right)$ lies on the line $y=\frac{1}{2}x+\frac{11}{2}$.

$(1)$ Find the general formula for the sequence $\{a_n\}$;

$(2)$ Let $b_n=\frac{3}{(2a_n-11)(2a_{n+1}-11)}$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $T_n$, and find the maximum positive integer value of $k$ such that the inequality $T_n > \frac{k}{20}$ holds for all $n \in \mathbb{N}^*$.

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