
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 parabola equation \( y = ax^2 + bx + c \) (\( a \neq 0 \)), there is an inscribed circle with radius \( \frac{1}{|a|} \) touching the parabola, denoted as \( \odot C_{1} \). A circle \( \odot C_{2} \) is drawn, which is externally tangent to \( \odot C_{1} \) and also touches the parabola internally. Then, a circle \( \odot C_{3} \) is drawn, which is externally tangent to \( \odot C_{2} \) and also touches the parabola internally, and so on, forming a sequence of circles \( C_{1}, C_{2}, C_{3}, \cdots \). Find the sum of the areas of the first \( n \) circles.

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