
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 the quadratic function $y = a(a+1)x^2 - (2a+1)x + 1$, where $a$ is a positive integer.
(1) If the graph of function $y$ intersects the x-axis at points A and B, find the length of segment AB;
(2) If $a$ takes the values 1, 2, ..., 2005 in turn, the graph of function $y$ intersects the x-axis, resulting in 2005 segments denoted as $A_1B_1$, $A_2B_2$, ..., $A_{2005}B_{2005}$. Try to find the sum of the lengths of these 2005 segments.

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