
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 $[x]$ denote the greatest integer less than or equal to the real number $x$,
\[
S = \left[\frac{1}{1}\right] + \left[\frac{2}{1}\right] + \left[\frac{1}{2}\right] + \left[\frac{2}{2}\right] + \left[\frac{3}{2}\right] + \left[\frac{4}{2}\right] + \left[\frac{1}{3}\right] + \left[\frac{2}{3}\right] + \left[\frac{3}{3}\right] + \left[\frac{4}{3}\right] + \left[\frac{5}{3}\right] + \left[\frac{6}{3}\right] + \cdots
\]

until the 2016th term. Each segment with denominator $k$ has $2k$ terms $\left[\frac{1}{k}\right], \left[\frac{2}{k}\right], \cdots, \left[\frac{2k}{k}\right]$, though the last segment may have fewer than $2k$ terms. Determine the value of $S$.

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