
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  $S$  be the set of integers between  $1$  and  $2^{40}$  whose binary expansions have exactly two  $1$ 's. If a number is chosen at random from  $S$ , the probability that it is divisible by  $9$  is  $p/q$ , where  $p$  and  $q$  are relatively prime positive integers. Find  $p+q$ .

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