
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 $\left\{a_n\right\}$ whose sum of the first $n$ terms is $S_n$, and the point $(n,S_n)$ lies on the curve $y= \frac{1}{2}x^2+ \frac{5}{2}x$. The sequence $\left\{b_n\right\}$ satisfies $b_n+b_{n+2}=2b_{n+1}$, $b_4=11$, and the sum of the first $5$ terms of $\left\{b_n\right\}$ is $45$.

(I) Find the general formula for $\left\{a_n\right\}$ and $\left\{b_n\right\}$;

(II) Let $C_n= \frac{1}{(2a_n-3)(2b_n-8)}$, and the sum of the first $n$ terms of $\left\{C_n\right\}$ is $T_n$. Find the maximum positive integer value of $k$ for which the inequality $T_n > \frac{k}{54}$ always holds.

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