
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]


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Given a sequence \\(\{a_n\}\\) with the sum of the first \\(n\\) terms denoted as \\(S_n\\), and it satisfies \\(2S_n = n^2 - n\\) (where \\(n \in \mathbb{N}^*\)).  
(Ⅰ) Find the general formula for the sequence \\(\{a_n\}\\).  
(Ⅱ) Let \\(b_n = \begin{cases} 2^{a_n}, & n=2k-1 \\ \frac{1}{n^2+2n}, & n=2k \end{cases} (k \in \mathbb{N}^*)\\), find the sum of the first \\(n\\) terms of the sequence \\(\{b_n\}\\), denoted as \\(T_n\\).

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