
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]


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Given a sequence $\left\{ a_n \right\}$ where each term is positive, and the sum of the first $n$ terms is $S_n$, it is known that $2$, $a_n$, and $S_n$ form an arithmetic sequence.

(Ⅰ) Find the general formula for the sequence $\left\{ a_n \right\}$;

(Ⅱ) Let $a_n^2 = \left(\frac{1}{2}\right)^{b_n}$, 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}
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