
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 the sequence $\{a_{n}\}$, with the "average reciprocal" of the first $n$ terms equal to $\frac{1}{{3n+1}}$, and the sequence $\{b_n\}$ defined by ${b_n}=\frac{{a_n+2}}{6}$, calculate the value of $\frac{1}{{b_1}{b_2}}+\frac{1}{{b_2}{b_3}}+\ldots+\frac{1}{{b_9}{b_{10}}}$.

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