
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]


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The sequence $\{a_n\}$ is an arithmetic sequence with a non-zero common difference, and $S_n$ is the sum of its first $n$ terms. It is known that $a_2+a_3+a_5=20$, and $a_2$, $a_4$, $a_8$ form a geometric sequence. Let $M= \frac{1}{S_1} + \frac{1}{S_2} + \ldots + \frac{1}{S_n}$.

(1) Find $M$;  
(2) The sum of the first $n$ terms of the sequence $\{b_n\}$ is $T_n$, and it is known that $T_n=2(b_n-1)$. Compare the size of $T_n$ and $M+1$.

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