
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 geometric sequence $\{a_{n}\}$ with the sum of the first $n$ terms as $S_{n}$, where $a_{n} \gt 0$ and $a_{1}a_{3}=36$, $a_{3}+a_{4}=9(a_{1}+a_{2})$.<br/>$(1)$ Find the general term formula for the sequence $\{a_{n}\}$;<br/>$(2)$ If $S_{n}+1=3^{b_{n}}$, find the sequences $\{b_{n}\}$ and $\{a_{n}b_{n}\}$, and the sum of the first $n$ terms $T_{n}$ of the latter sequence;<br/>$(3)$ If $C_{n}=\frac{1}{b_{n}b_{n+1}}$, find the sum $M_{n}$ of the first $n$ terms of the sequence $\{C_{n}\}$.

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