
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 $f(x) = \lfloor x \rfloor (2013^{x-\lfloor x\rfloor} - 1)$ for every real number $x$. Determine the sum of the lengths of all intervals $x$ such that $1 \leq x < 2013$ and $f(x) \leq 1$.
A) $0$
B) $\frac{\log 2014}{\log 2013}$
C) $1$
D) $\frac{\log 2015}{\log 2014}$
E) $2013^{\frac{1}{2013}}$

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