
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 $a_n = \int_{0}^{n}(2x+1)dx$, where $(n \in \mathbb{N}^*)$, the sum of the first $n$ terms of the sequence $\{\frac{1}{a_n}\}$ is $S_n$, and the general term of the sequence $\{b_n\}$ is $b_n = n - 8$. Determine the minimum value of $b_nS_n$.

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