
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 the sequences $\{a_{n}\}$, $\{b_{n}\}$, and $S_{n}$ is the sum of the first $n$ terms of the sequence $\{a_{n}\}$. It is known that for any $n\in \mathbb{N}^{*}$, $3a_{n}=2S_{n}+3$. The sequence $\{b_{n}\}$ is an arithmetic sequence, $b_{1}=\log _{3}a_{1}$, and $b_{2}+5$, $b_{4}+1$, $b_{6}-3$ form a geometric sequence.  
$(1)$ Find the general term formulas for the sequences $\{a_{n}\}$ and $\{b_{n}\}$;  
$(2)$ Let ${d}_{n}=\frac{{b}_{n+2}-1}{{b}_{n}{b}_{n+1}{a}_{n}},n\in \mathbb{N}^{*}$. Find the sum of the first $n$ terms of the sequence $\{d_{n}\}$ denoted as $T_{n}$;  
$(3)$ Let ${c}_{n}=\left\{\begin{array}{l}{a}_{n}, \text{ for odd } n\\{b}_{\frac{n}{2}}, \text{ for even } n\end{array}\right.$. Find $\sum_{k=1}^{2n}{{c}_{k}}{c}_{k+1}$.

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