
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
Let the sequence $\{a_{n}\}\left(n\in N*\right)$ be an arithmetic sequence with a non-zero common difference, satisfying $a_{3}+a_{6}=a_{9}$ and $a_{5}+a_{7}^{2}=6a_{9}$; the sequence $\{b_{n}\}\left(n\in N*\right)$ has a sum of the first $n$ terms as $S_{n}$, and satisfies $4S_{n}+2b_{n}=3$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Insert one number $x_{11}$ between $b_{1}$ and $b_{2}$ to form an arithmetic sequence; insert two numbers $x_{21}$ and $x_{22}$ between $b_{2}$ and $b_{3}$ to form an arithmetic sequence; and so on. Let $T_{n}=x_{11}+x_{21}+x_{22}+\ldots +x_{n1}+x_{n2}+\ldots +x_{nn}$.  
$(i)$ Find $T_{n}$.  
$(ii)$ Determine if there exist positive integers $m$ and $n$ such that $T_{n}=\frac{{{a_{m+1}}}}{{2{a_m}}}$ holds. If they exist, find all positive integer pairs $\left(m,n\right)$; if not, explain why.

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