
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 $\dfrac{n}{p_{1}+p_{2}+\ldots+p_{n}}$ as the "average reciprocal" of $n$ positive numbers $p_{1}$, $p_{2}$, $\ldots$, $p_{n}$, if $\dfrac{n}{a_{1}+a_{2}+\ldots+a_{n}}= \dfrac{1}{2n+1}$ and $b_{n}= \dfrac{a_{n}+1}{4}$, calculate $\dfrac{1}{b_{1}b_{2}}+ \dfrac{1}{b_{2}b_{3}}+\ldots+ \dfrac{1}{b_{10}b_{11}}$.

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