
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
1007 distinct potatoes are chosen independently and randomly from a box of 2016 potatoes numbered  $1, 2, \dots, 2016$ , with  $p$  being the smallest chosen potato. Then, potatoes are drawn one at a time from the remaining 1009 until the first one with value  $q < p$  is drawn. If no such  $q$  exists, let  $S = 1$ . Otherwise, let  $S = pq$ . Then given that the expected value of  $S$  can be expressed as simplified fraction  $\tfrac{m}{n}$ , find  $m+n$ .

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