
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 pairs  $(n, q)$ , where  $n$  is a positive integer and  $q$  a non-integer rational number with  $0 < q < 2000$ , that satisfy  $\{q^2\}=\left\{\frac{n!}{2000}\right\}$ 

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