
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 positive integers \(a_{1}, a_{2}, \cdots, a_{2018}\) satisfying
\[ a_{1} < a_{2} < \cdots < a_{2018}, \]

for \(i = 1, 2, \cdots, 2018\), let \(b_{i}\) be the number of positive integers among \(a_{1}, a_{2}, \cdots, a_{2018}\) that are less than or equal to \(i\). Then, find the value of
\[ \frac{\sum_{k=1}^{2018} a_{k} + \sum_{k=1}^{a_{2018}} b_{k}}{a_{2018} + 1}. \]

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