
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 $20$ balls is tossed independently and at random into one of $5$ bins, calculate the probability $p$ that one bin ends up with $2$ balls, two bins each with $4$ balls, one with $3$ balls, and the other one with $7$ balls, and then calculate the probability $q$ that two bins each end up with $3$ balls, and the other three bins each with $4$ balls, and finally determine the value of $\frac{p}{q}$.

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