
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 rectangular coordinate plane $xOy$, there are 2018 circles $\odot A_{1}, \odot A_{2}, \cdots, \odot A_{2018}$, where the center of $\odot A_{k} (k=1, 2, \cdots, 2018)$ is $A_{k}\left(a_{k}, \frac{1}{4} a_{k}^{2}\right)$, the radius is $\frac{1}{4} a_{k}^{2}$, $a_{1} > a_{2} > \cdots > a_{2108} = \frac{1}{2018}$, and $\odot A_{k+1}$ is externally tangent to $\odot A_{k}$ $(k=1, 2, \cdots, 2017)$. Find the value of $a_{1}$.

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