
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 $\{a_n\}$ where all terms are positive, the sum of the first $n$ terms is denoted as $S_n$, and $2, a_n, S_n$ form an arithmetic sequence.

(1) Find the general term formula for the sequence $\{a_n\}$.

(2) If $a_n^2 = \left(\frac{1}{2}\right)^{b_n}$, find the value of $\frac{1}{b_1b_2} + \frac{1}{b_2b_3} + \cdots + \frac{1}{b_nb_{n+1}}$.

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