
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
(1) Given the sequence $\{a_n\}$ with the sum of the first $n$ terms $S_n=\frac{3(3^n+1)}{2}$, find the general formula for $\{a_n\}$.

(2) Given an arithmetic sequence $\{a_n\}$ with the sum of the first $n$ terms as $S_n$, where $n\in\mathbb{N}^*$, and $a_3+a_6=4$, $S_5=-5$.

   (i) Find $a_n$;
   
   (ii) If $T_n=|a_1|+|a_2|+|a_3|+\cdots+|a_n|$, find the value of $T_5$ and the expression for $T_n$.

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