
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
Given an arithmetic sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, $a_{3}=5$, $S_{6}-S_{3}=27$, and a sequence $\{b_{n}\}$ with the product of the first $n$ terms as ${T}_{n}={3}^{\frac{n(n+1)}{2}}$.  
$(1)$ Find the general formulas for $\{a_{n}\}$ and $\{b_{n}\}$.  
$(2)$ Let ${c}_{n}=\frac{{a}_{n}{b}_{n}}{{n}^{2}+n}$, determine the sum of the first $n$ terms of the sequence $\{c_{n}\}$, denoted as $Q_{n}$.

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