
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 terms of the geometric sequence $\{a_n\}$ are all positive, and $a_1 + 2a_2 = 1$, and $a_3^2 = 4a_2 \cdot a_6$.  
(1) Find the general formula for the sequence $\{a_n\}$.  
(2) Let $b_n = \log_2 a_1 + \log_2 a_2 + \log_2 a_3 + \ldots + \log_2 a_n$, find the sum of the first $n$ terms of the sequence $\left\{ \frac{1}{b_n} \right\}$.  
(3) Let $c_n = \frac{b_n \cdot a_n}{n}$, find the sum of the first $n$ terms of the sequence $\{c_n\}$.

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