
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\}\) where \\(a_2=2\\), and the sum of the first \\(n\\) terms \\(S_n\\) satisfies: \\(S_n= \frac{n(a_n-a_1)}{2} (n\in\mathbb{N}^*).\\)  
(Ⅰ) Find the general formula for the sequence \\(\{a_n\}\);  
(Ⅱ) If \\(b_n=n\cdot2^{a_n}\\), find the sum of the first \\(n\\) terms of the sequence \\(\{b_n\}\), \\(T_n.\\)

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