
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
Let  $a,b,c,d$  be positive integers such that the number of pairs  $(x,y) \in (0,1)^2$  such that both  $ax+by$  and  $cx+dy$  are integers is equal with 2004. If  $\gcd (a,c)=6$  find  $\gcd (b,d)$ .

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