
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 $\{a_{n}\}$ be a sequence with the sum of its first $n$ terms denoted as $S_{n}$. It is known that $a_{1}=3$, and $\left\{{\frac{{{S_n}}}{n}}\right\}$ forms an arithmetic sequence with a common difference of $2$.  
$(1)$ Find the general formula for $\{a_{n}\}$.  
$(2)$ Let ${b_n}=\frac{1}{{{a_n}{a_{n+1}}}$. Determine the sum of the first $n$ terms of the sequence $\{b_{n}\}$ denoted as $T_{n}$.

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