
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
For the sequence $\{a_n\}$, with $a_1 = 1$, $a_n$ and $a_{n+1}$ are the two roots of the quadratic equation $x^2 - (2n+1)x + \frac{1}{b_n} = 0$. Find the sum of the first $n$ terms of the sequence $\{b_n\}$, denoted as $S_n$.

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