
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 \[f(x) = \lfloor x\rfloor (2013^{x-\lfloor x \rfloor}-1).\] Determine the sum of the lengths of all intervals for $x$ such that $1 \leq x < 2013$ and $f(x) \geq 1$.
A) 2013
B) 2012
C) 2011
D) 2010
E) 2009

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