
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 an arithmetic sequence $\left\{ a_n \right\}$ with a common difference $d$ not equal to $0$, where $a_1=2$, and $a_2+1$, $a_4+1$, $a_8+1$ form a geometric sequence.

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

(2) Suppose there is a sequence $\left\{ b_n \right\}$ defined by $b_n=\frac{3}{a_n}$. Find the value of the positive integer $n$ for which the equation $b_1b_2 + b_2b_3 + \cdots + b_nb_{n+1} = \frac{45}{32}$ holds.

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