
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  $a/b$  be the probability that a randomly chosen positive divisor of  $12^{2007}$  is also a divisor of  $12^{2000}$ , where  $a$  and  $b$  are relatively prime positive integers.  Find the remainder when  $a+b$  is divided by  $2007$ .

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