
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 $\{a_n\}$ with a common difference 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 formula for the sequence $\{a_n\}$.  
$(2)$ Let the sequence $\{b_n\}$ satisfy $b_n= \frac {3}{a_n}$, find the value of the positive integer $n$ that satisfies the equation $b_1b_2+b_2b_3+\ldots+b_nb_{n+1}= \frac {45}{32}$.

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