
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 \\(\left\{ a_n \right\}\\) with the sum of the first \\(n\\) terms denoted as \\(S_n\\), and \\(S_n = n^2 - 4n + 4, (n \in N^*)\\),

\\((1)\\) Find the general formula for the sequence \\(\left\{ a_n \right\}\\);

\\((2)\\) For the sequence \\(\left\{ b_n \right\}\\), let \\(b_n = \begin{cases} & 1, & n=1 \\ & \frac{a_n + 5}{2}, & n \geqslant 2 \end{cases}\\), and \\(T_n = 2b_1 + 2^2b_2 + 2^3b_3 + \cdots + 2^nb_n\\), find \\(T_n\\).

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