
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 any \( n \in \mathbf{N}^{*} \), the parabola \( y = (n^{2} + n)x^{2} - (2n + 1)x + 1 \) intersects the \( x \)-axis at points \( A_{n} \) and \( B_{n} \). Find the sum \( \left|A_{1} B_{1}\right| + \left|A_{2} B_{2}\right| + \cdots + \left|A_{2009} B_{2009}\right| \).

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