
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 \(\lfloor x\rfloor\) denote the greatest integer not exceeding \(x\), and define the function \[ f(x) = \lfloor x\rfloor(2013^{x-\lfloor x\rfloor}-1). \] Find the sum of the lengths of all intervals for \(x\) such that \(1 \leq x < 2013\) and \(f(x) \leq 1\).

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