
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
Determine the smallest positive integer \( q \) with the following property: for every integer \( m \) with \( 1 \leq m \leq 1006 \), there exists an integer \( n \) such that

\[ 
\frac{m}{1007} q < n < \frac{m+1}{1008} q .
\]

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