
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
The sequence \(\left\{a_{n}\right\}\) satisfies: \(a_{1}=1\), and for each \(n \in \mathbf{N}^{*}\), \(a_{n}\) and \(a_{n+1}\) are the two roots of the equation \(x^{2}+3nx+b_{n}=0\). Find \(\sum_{k=1}^{20} b_{k}\).

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