
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 \(\left\{a_{n}\right\}\) be an arithmetic sequence with a common difference that is non-zero, and it satisfies the conditions \(a_{3} + a_{6} = a_{9}\) and \(a_{5} + a_{7}^{2} = 6a_{9}\). Denote the sum of the first \(n\) terms of the sequence \(\left\{b_{n}\right\}\) by \(S_{n}\), and \(4S_{n} + 2b_{n} = 3\). For any \(i \in \mathbf{Z}_{+}\), insert \(i\) numbers \(x_{i1}, x_{i2}, \cdots, x_{ii}\) between \(b_{i}\) and \(b_{i+1}\) such that \(b_{i}, x_{i1}, x_{i2}, \cdots, x_{ii}, b_{i+1}\) forms an arithmetic sequence. Let \(T_{n} = \sum_{i=1}^{n} \sum_{j=1}^{i} x_{i j}\). Does there exist positive integers \(m\) and \(n\) such that \(T_{n} = \frac{a_{m+1}}{2a_{m}}\)? If so, find all such pairs of positive integers \((m, n)\); if not, explain why.

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