
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 of positive terms with the sum of the first $n$ terms denoted as $S_{n}$. It is known that the points $({a}_{n},6{S}_{n})$ for $n \in \mathbb{N}^{*}$ lie on the graph of the function $f\left(x\right)=\left(x+1\right)\left(x+2\right)$, and $a_{1} \gt 1$. The sequence $\{b_{n}\}$ satisfies $b_{n+1}=3b_{n}$, with $b_{2}=a_{3}+1$.  
$(1)$ Find the general formulas for sequences $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Let ${c}_{n}=\frac{{a}_{n}}{{b}_{n+1}}+\frac{{a}_{1}-1}{{b}_{n}}$. Find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted as $T_{n}$.

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