
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
For every real number $x$, let $\lfloor x\rfloor$ denote the greatest integer not exceeding $x$, and let \[f(x)=\lfloor x\rfloor(2013^{x-\lfloor x\rfloor}-1).\]  The set of all numbers $x$ such that $1\leq x<2013$ and $f(x)\leq 1$ is a union of disjoint intervals.  What is the sum of the lengths of those intervals?
A) \( 1 \)
B) \( \frac{1}{2013} \)
C) \( \frac{\log 2013}{\log 2012} \)
D) \( 2013 \)

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