
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 an arithmetic sequence with a common difference of $d$, and $d \gt 1$. Define $b_{n}=\frac{{n}^{2}+n}{{a}_{n}}$, and let $S_{n}$ and $T_{n}$ be the sums of the first $n$ terms of the sequences $\{a_{n}\}$ and $\{b_{n}\}$, respectively.  
$(1)$ If $3a_{2}=3a_{1}+a_{3}$ and $S_{3}+T_{3}=21$, find the general formula for $\{a_{n}\}$.  
$(2)$ If $\{b_{n}\}$ is an arithmetic sequence and $S_{99}-T_{99}=99$, find $d$.

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