
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
Consider a sequence \( s_1, s_2, \ldots, s_{50} \) of distinct real numbers in random order. A bubble pass through this sequence compares and potentially swaps adjacent elements if the right one is smaller than the left one. Suppose after one bubble pass, the element initially at position \( s_{25} \) has moved to position \( s_{40} \). Determine the numerical sum \( p + q \) where \( \frac{p}{q} \), in lowest terms, represents the probability of this event.

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