
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 $r_1, r_2, \dots, r_{50}$ of 50 distinct real numbers in random order. Perform one bubble pass as described: compare each pair $(r_k, r_{k+1})$ for $k=1$ to $49$, and exchange them if $r_{k+1} < r_k$. Given this, compute the probability in lowest terms that the number originally at position $r_{25}$ moves to the $35^{th}$ position after one bubble pass. Provide the sum $p + q$ where the probability is $p/q$.

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