
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
Call a quadruple of positive integers  $(a, b, c, d)$  fruitful if there are infinitely many integers  $m$  such that  $\text{gcd} (am + b, cm + d) = 2019$ . Find all possible values of  $|ad-bc|$  over fruitful quadruples  $(a, b, c, d)$ .

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