
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
In the Cartesian coordinate plane \( xOy \), there are infinitely many circles inside the parabola \( y = ax^2 \) (where \( a \) is a constant). The centers of the circles \( O_1, O_2, \cdots \) are all on the y-axis. For each integer \( n > 1 \), the circle \( \odot O_n \) is tangent to the parabola and also externally tangent to the circle \( \odot O_{n-1} \). If the radius of \( \odot O_1 \) is \( \frac{1}{2a} \) and it passes through the vertex of the parabola, find the radius of \( \odot O_{2016} \).

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