
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 terms are all non-zero and its sum of the first \\(n\\) terms \\(S_n\\), and it satisfies \\(4S_n = (2n+1)a_n + 1\\), the sequence \\(\{b_n\}\) satisfies \\(b_1 = 1\\), \\(b_{n+1} = 2b_n + 1\\).  
\\((I)\\) Find the general formula for the sequences \\(\{a_n\}\) and \\(\{b_n\}\);  
\\((II)\\) Let \\(c_n = a_n(b_n + 1)\\), find the sum of the first \\(n\\) terms of the sequence \\(\{c_n\}\), \\(T_n\\).

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