
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
In the arithmetic sequence $\left\{a_n\right\}$, $a_3+a_4=12$ and the common difference $d=2$. Let the sum of the first $n$ terms of the sequence $\left\{a_{2n-1}\right\}$ be $S_n$.

(1) Find $S_n$;

(2) Let the sum of the first $n$ terms of the sequence $\left\{ \frac{n}{a_{n+1}S_n}\right\}$ be $T_n$. If $a_2$, $a_5$, $a_m$ form a geometric sequence, find the value of $T_m$.

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