
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 $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, and $S_n$ is the sum of its first $n$ terms, $S_3=9$, and $a_2, a_5, a_{14}$ form a geometric sequence, the sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n=\frac{3^{n+1}-3}{2}$.
1. Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.
2. If $c_n=\frac{a_n^2+8\log_3 b_n}{a_{n+1} b_n}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $M$.

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