
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 $15$ balls is tossed independently and at random into one of the $4$ bins, find the probability that one bin ends up with $3$ balls, another with $6$ balls, and the other two with $3$ balls each, and then determine the probability that every bin ends up with $3$ balls, $2$ balls, $3$ balls, and $7$ balls respectively, and calculate the ratio of these probabilities.

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