
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 first $n$ terms have a sum of $S_n$, and the point $(n, \frac{S_n}{n})$ lies on the line $y = \frac{1}{2}x + \frac{11}{2}$. Another sequence ${b_n}$ satisfies $b_{n+2} - 2b_{n+1} + b_n = 0$ ($n \in \mathbb{N}^*$), and $b_3 = 11$, with the sum of the first 9 terms being 153.
(I) Find the general term formulas for the sequences ${a_n}$ and ${b_n}$;
(II) 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 $k$ such that the inequality $T_n > \frac{k}{57}$ holds for all $n \in \mathbb{N}^*$.

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