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

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

(2) Let $c_n = a_n - kb_n$, and the sum of the first $n$ terms of the sequence $\{c_n\}$ be $T_n$. Determine the maximum value of positive integer $k$ such that the inequality $T_n > 0$ holds for all $n\in \mathbb{N^*}$.

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