
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
How many of the first $1000$ positive integers can be expressed in the form
\[
\lfloor 3x \rfloor + \lfloor 6x \rfloor + \lfloor 9x \rfloor + \lfloor 12x \rfloor
\]
where $x$ is a real number, and $\lfloor z \rfloor$ denotes the greatest integer less than or equal to $z$?

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