
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
In an arithmetic sequence $\{a_n\}$, it is given that $a_{16}+a_{17}+a_{18} = a_{9} = -36$, and $S_n$ denotes the sum of the first $n$ terms.

(1) Find the minimum value of $S_n$ and determine the value of $n$ when $S_n$ is at its minimum.

(2) Consider the sequence $\{b_n\} = \frac{3}{a_{n}+66}$, find the sum of the first $n$ terms of the product sequence $\{b_n b_{n+1}\}$.

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