
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
Each of the $25$ balls is tossed independently and at random into one of $5$ bins. Let $r$ be the probability that one bin ends up with $6$ balls, another with $7$ balls, and the other three with $4$ balls each. Let $s$ be the probability that one bin ends up with $5$ balls and the other four with $5$ balls each. Compute the ratio $\frac{r}{s}$.
**A)** 5  
**B)** $\frac{10}{3}$  
**C)** $\frac{10 \cdot \binom{25}{6}\binom{19}{7}\binom{12}{4}\binom{8}{4}\binom{4}{4}}{\binom{25}{5}\binom{20}{5}\binom{15}{5}\binom{10}{5}\binom{5}{5}}$  
**D)** 15

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