
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
In the sequence $\{a_{n}\}$, ${a_n}=1+\frac{1}{{2n-9}}$ ($n∈{N^*},a∈R,a≠0$), the values of its maximum term and minimum term are the values of $b_{2}-1$ and $b_{3}-9$ in the geometric sequence $\{b_{n}\}$, respectively.  
$(1)$ Find the general formula of the sequence $\{b_{n}\}$;  
$(2)$ Given the sequence $\{c_{n}\}$, $c_{n}=b_{n}\cdot \log _{3}(b_{n})$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted as $M_{n}$.

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