
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
Let \(a_{1}, a_{2}, \cdots, a_{n}\) be an increasing sequence of positive integers. For a positive integer \(m\), define
\[b_{m}=\min \left\{n \mid a_{n} \geq m\right\} (m=1,2, \cdots),\]
that is, \(b_{m}\) is the smallest index \(n\) such that \(a_{n} \geq m\). Given \(a_{20}=2019\), find the maximum value of \(S=\sum_{i=1}^{20} a_{i}+\sum_{i=1}^{2019} b_{i}\).

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