
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 a Cartesian coordinate system $xOy$, there are infinitely many circles within the parabola $y = ax^2$ (where $a$ is a constant). The centers of these 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 externally tangent to the circle $\odot O_{n-1}$. If the radius of the circle $\odot O_1$ is $\frac{1}{2a}$ and it passes through the vertex of the parabola, find the radius of the circle $\odot O_{2016}$.

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