
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 a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n$, and $a_1=2$, $2S_n=(n+1)^2a_n-n^2a_{n+1}$, there exists a sequence $\{b_n\}$ such that $b_1=a_1$ and $nb_{n+1}=a_nb_n$.  
(i) Find the general formula for the sequences $\{a_n\}$ and $\{b_n\}$.  
(ii) If there is a sequence $\{c_n\}$ such that $c_n=a_n+b_n$ for $n\in\mathbb{N^*}$, find the sum of the first $n$ terms of sequence $\{c_n\}$, denoted as $T_n$.

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