
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 $S_n$, the point $(n, \frac{S_{n}}{n})$ lies on the line $y=\frac{1}{2}x+\frac{11}{2}$. Another sequence $\{b_n\}$ satisfies the relation $b_{n+2} - 2b_{n+1} + b_n = 0$ ($n\in \mathbb{N}^*$), and it is given that $b_3=11$ and the sum of the first 9 terms is 153.  
(1) Find the general formulas 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 greatest positive integer $k$ such that the inequality $T_n > \frac{k}{57}$ holds for all $n \in \mathbb{N}^*$.  
(3) Suppose $n \in \mathbb{N}^*$ and $f(n)= 
\begin{cases} 
a_{n}, & \text{if } n \text{ is odd} \\
b_{n}, & \text{if } n \text{ is even}
\end{cases}$. Is there an $m \in \mathbb{N}^*$ such that $f(m+15)=5f(m)$? If yes, find the value of $m$; if not, explain why.

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