
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
Find the number of positive even integers $n \le 1000$ that can be expressed in the form
\[
\lfloor x \rfloor + \lfloor 2x \rfloor + \lfloor 4x \rfloor = n
\]
for some real number $x$.

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