
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 positive integers \( n \) less than 2007 can we find such that \(\left\lfloor \frac{n}{2} \right\rfloor + \left\lfloor \frac{n}{3} \right\rfloor + \left\lfloor \frac{n}{6} \right\rfloor = n\), where \(\left\lfloor x \right\rfloor\) is the greatest integer less than or equal to \( x \)? (For example, \(\left\lfloor 2.5 \right\rfloor = 2\), \(\left\lfloor 5 \right\rfloor = 5\), \(\left\lfloor -2.5 \right\rfloor = -3\), etc.)

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