
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 that the sequence $\{a_n\}$ is a non-zero arithmetic sequence with a common difference $d$, and $S_n$ is the sum of its first $n$ terms, satisfying $a_n^2 = S_{2n-1}$ for $n\in\mathbb{N}^*$. The sequence $\{b_n\}$ is defined by $b_n = \dfrac{1}{a_n a_{n+1}}$, and let $T_n$ be the sum of the first $n$ terms of $\{b_n\}$.
(1) Find the general term formula for $\{a_n\}$ and $T_n$;
(2) Is there a positive integer $m$, $n(1 < m < n)$, such that $T_1$, $T_m$, and $T_n$ form a geometric sequence? If it exists, find all values of $m$ and $n$; if not, explain why.

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