
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]


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Given a sequence $\{a_n\}$ with the sum of its first $n$ terms denoted as $S_n$, 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 its first $9$ terms being $153$.

$(1)$ Find the general formula for 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 $n\in\mathbb{N}^*$, and define $f(n)=\begin{cases}a_n, & \text{if } n \text{ is odd} \\ b_n, & \text{if } n \text{ is even}\end{cases}$. Is there an $m\in \mathbb{N}^*$ such that $f(m+15)=5f(m)$ holds? If it exists, find the value of $m$; if not, explain why.

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