
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
Given that each of the 25 balls is tossed independently and at random into one of the 5 bins, let $p$ be the probability that some bin ends up with $3$ balls, another with $7$ balls, and the other three with $5$ balls each, and let $q$ be the probability that every bin ends up with $5$ balls; find $\frac{p}{q}$.

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