
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 \( \{a_n\} \) satisfies: \( a_1 = 1 \), and for each \( n \in \mathbf{N}^{*} \), \( a_n \) and \( a_{n+1} \) are the roots of the equation \( x^2 + 3n x + b_n = 0 \). Find the value of \( \sum_{k=1}^{20} b_k \).

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