
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 a corner of a stadium, there are a total of $20$ rows of stands, and the arrangement of seats is as follows: the first row has $2$ seats, and starting from the second row, each row has one more seat than the previous row. Let $a_n$ represent the number of seats in the $n$-th row.  
$(1)$ Determine the total number of seats in these stands;  
$(2)$ Calculate the sum of the first $20$ terms, $S_{20}$, of the sequence $\left\{ \frac{a_n}{n(n+1)^2} \right\}$.

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