
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 $a_{n}= \int_{0}^{n}(2x+1)dx$, the sum of the first $n$ terms of the sequence $\{\frac{1}{a_{n}}\}$ is $S_{n}$. The general term of the sequence $\{b_{n}\}$ is $b_{n}=n-8$. The minimum value of $b_{n}S_{n}$ is ____.

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