
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
Suppose a sequence $s_1, s_2, \dots, s_{50}$ of 50 distinct real numbers is randomly ordered. A bubble pass procedure is used to sort these numbers in ascending order as described previously. Calculate the probability that the number initially at position $s_{10}$ will end up at position $s_{15}$ after one bubble pass, if each position change is contingent upon the adjacent value comparisons as per a bubble sort's mechanism. Express your answer as a sum of numerator and denominator in its simplest fractional form.

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