
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 define the function
\[
f(x) = \lfloor x \rfloor (2020^{x - \lfloor x \rfloor} - 1).
\]
Determine the sum of the lengths of intervals for $x$ such that $1 \leq x < 2020$ and $f(x) \leq 1$.
A) $1$
B) $\frac{\log 2021}{\log 2020}$
C) $\frac{2019}{2020}$
D) $2020^{\frac{1}{2020}}$
E) $0$

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