
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 \(\lfloor x\rfloor\) be the greatest integer smaller than or equal to \(x\). How many solutions are there to the equation \(x^{3} - \lfloor x^{3} \rfloor = (x - \lfloor x \rfloor)^{3}\) on the interval \([1, 20]\)?

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