
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 $S_n= \frac{1}{2}n^2+ \frac{11}{2}n$, the sequence $\{b_n\}$ satisfies $b_{n+2}-2b_{n+1}+b_{n}=0$ ($n\in{N^*}$), and $b_3=11$, the sum of the first $9$ terms is $153$.

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

(2) Let $c_n= \frac{3}{(2a_n-11)(2b_n-1)}$, the sum of the first $n$ terms of the sequence $\{c_n\}$ is $T_n$, 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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