
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 $\{a_n\}$ have a sum of the first $n$ terms denoted by $S_n$, and $\left\{ \frac{S_n}{n} \right\}$ forms an arithmetic sequence. Given that $a_1=1$ and $\frac{S_2}{2} + \frac{S_3}{3} + \frac{S_4}{4} = 6$.

$(1)$ Find the general formula for $\{a_n\}$;  
$(2)$ If $b_n= \frac{a_{n+1}}{a_{n+2}} + \frac{a_{n+2}}{a_{n+1}} - 2$, find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted by $T_n$.

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