
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 $(n, \frac{S_n}{n})$ lies on the line $y= \frac{1}{2}x+ \frac{11}{2}$. The sequence $\{b_n\}$ satisfies $b_{n+2}-2b_{n+1}+b_n=0$ $(n\in{N}^*)$, and $b_3=11$, with the sum of the first $9$ terms being $153$.

$(1)$ Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$;

$(2)$ Let $c_n= \frac{3}{(2a_n-11)(2b_n-1)}$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$. Find the maximum positive integer value of $k$ such that the inequality $T_n > \frac{k}{57}$ holds for all $n\in{N}^*$;

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