
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
The sequence $\{a_n\}$ is a positive geometric sequence, and it satisfies $a_1 + \frac{1}{2}a_2 = 4$, $a_3^2 = \frac{1}{4}a_2a_6$; Let the sum of the first $n$ terms of the positive sequence $\{b_n\}$ be $S_n$, and it satisfies $S_n = \frac{(b_n+1)^2}{4}$.  
(1) Find the general formula for $\{a_n\}$ and $\{b_n\}$.  
(2) Let $c_n = a_nb_n$, find the sum of the first $n$ terms of the sequence $\{c_n\}$, denoted as $T_n$.

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