
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
Consider the function defined for every real number \( x \) by \[ g(x) = \lfloor x \rfloor \left(2013^{x-\lfloor x\rfloor} - 1\right). \] Determine the sum of the lengths of all intervals of \( x \) such that \( 1 \leq x < 2015 \) and \( g(x) \leq 1 \).
A) \( \log_{2013}(2015) \)
B) \( \log_{2015}(2013) \)
C) \( \frac{\log 2016}{\log 2013} \)
D) \( \frac{2015}{2013} \)
E) \( 2013^{\frac{1}{2015}} \)

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