
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 that the sum of the first $n$ terms of the sequence $\{b_{n}\}$ is $S_{n}$, satisfying $2S_{n}=3(b_{n}-1)$, and in an arithmetic sequence $\{c_{n}\}$, $c_{1}=5$, $c_{1}+c_{2}+c_{3}=27$.  
$(1)$ Find the general formulas for $\{b_{n}\}$ and $\{c_{n}\}$.  
$(2)$ The common terms of sequences $\{b_{n}\}$ and $\{c_{n}\}$ are arranged in ascending order to form a new sequence $\{a_{n}\}$. Find the product $T_{20}$ of the first 20 terms of the sequence $\{a_{n}\}$.

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