
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 geometric sequence \\(\{a_n\}\) that satisfies: \\(a_1= \frac{1}{2}, 2a_3=a_2\\)  
\\((1)\\) Find the general formula for the sequence \\(\{a_n\}\);  
\\((2)\\) If the arithmetic sequence \\(\{b_n\}\) has a sum of the first \\(n\\) terms denoted as \\(S_n\\), satisfying \\(b_1=1\\) and \\(S_3=b_2+4\\), find the sum of the first \\(n\\) terms \\(T_n\\) for the sequence \\(\{a_n\cdot b_n\}\).

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