
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 that the sequence \\(\{a_n\}\) is an arithmetic sequence with a positive common difference, and the sequence \\(\{b_n\}\) is a geometric sequence, with \\(a_1=1\\), \\(b_1=2\\), \\(b_2-a_2=1\\), and \\(a_3+b_3=13\\)  
\\((\\)I\\()\\) Find the general formula for sequences \\(\{a_n\}\) and \\(\{b_n\}\)  
\\((\\)II\\()\\) Let \\(c_n=a_nb_n\\), 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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