
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
Set  $S$  contains exactly  $36$  elements in the form of  $2^m \cdot 5^n$  for integers  $ 0 \le m,n \le 5$ . Two distinct elements of  $S$  are randomly chosen. Given that the probability that their product is divisible by  $10^7$  is  $a/b$ , where  $a$  and  $b$  are relatively prime positive integers, find  $a +b$ .

*Proposed by Ada Tsui*

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