
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
Given a quadratic function $y = a(a+1)x^2 - (2a+1)x + 1$.

(1) When the graph of this function intersects the x-axis at two points, find the range of values for $a$.

(2) When $a$ is a positive integer, and the graph of this function intersects the x-axis at points $A$ and $B$, find the length of segment $AB$.

(3) If $a$ takes the values 1, 2, ..., 2010 successively, and the graph of the function intersects the x-axis, resulting in 2010 segments $A_1B_1, A_2B_2, ..., A_{2010}B_{2010}$, find the sum of their lengths.

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