
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 $\{a_n\}$ with the sum of its first $n$ terms being $S_n$, the point $(n, \frac{S_n}{n})$ lies on the line $y= \frac{1}{2}x+ \frac{11}{2}$. The 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 its first 9 terms being 153.
1. Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.
2. Find the sum of the first $n$ terms $T_n$ of the sequence $\{(a_n-5)\cdot 2^{a_n}\}$.
3. Let $n\in\mathbb{N}^*$, $f(n) = \begin{cases} a_n, & \text{if } n \text{ is odd} \\ b_n, & \text{if } n \text{ is even} \end{cases}$. Is there an $m\in\mathbb{N}^*$ such that $f(m+15) = 5f(m)$ holds? If so, find the value of $m$; if not, explain why.

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