
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 each real number \(x\), let \(\lfloor x\rfloor\) denote the greatest integer not exceeding \(x\), and define the function by \(f(x) = \lfloor x\rfloor(2024^{x-\lfloor x\rfloor} - 1)\). Determine the sum of the lengths of all intervals where \(1 \leq x < 2024\) and \(f(x) \leq 2\).
A) 1
B) \(\dfrac{\log 2025}{\log 2024}\)
C) \(\dfrac{\log 2024}{\log 2023}\)
D) \(\dfrac{2024}{2023}\)
E) 2025^{\frac{1}{2025}}

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