
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 all terms are positive, and the product of the first \\(n\\) terms is \\(T_n = 2^{\frac{n(n-1)}{2}}\\),  
(Ⅰ) Find the general formula for the sequence \\(\{a_n\}\).  
(Ⅱ) Calculate the sum: \\(S_n = a_1 + 2a_2 + \ldots + (n+2)a_{n+2} - (n+1)a_{n+3} - 1\\).

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