
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 $y=a(a+1)x^{2}-(2a+1)x+1$, when $a$ takes the values of $1, 2, 3, \dots, n$, the lengths of the line segments intercepted by the parabola on the $x$-axis are $d_{1}, d_{2}, d_{3}, \dots, d_{n}$ respectively. Find the limit $\lim\limits_{n→∞}(d_{1}+d_{2}+⋯+d_{n})$.

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