
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 denoted as $S_n$, $a_1=3$, and $3S_1$, $2S_2$, $S_3$ form an arithmetic sequence.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $b_n=\log_3a_n$, calculate the value of $T_n=b_1b_2-b_2b_3+b_3b_4-b_4b_5+\ldots+b_{2n-1}b_{2n}-b_{2n}b_{2n+1}$.

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