
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
The sum of the first n terms of the sequence $\{a_n\}$ is $S_n$.  
(1) When $\{a_n\}$ is a geometric sequence, $a_1=1$, and $\frac{1}{a_1}$, $\frac{1}{a_3}$, $\frac{1}{a_4}-1$ form an arithmetic sequence, find $a_n$;  
(2) If $\{a_n\}$ is an arithmetic sequence, and $S_1+a_2=3$, $S_2+a_3=6$, find the sum: $T_n= \frac{1}{S_1}+ \frac{1}{S_2}+ \frac{1}{S_3}+…+ \frac{1}{S_n}$.

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