
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
In a random ball distribution, each of the $30$ balls is tossed independently into one of the $6$ bins. Let \( p' \) be the probability that one bin ends up with $6$ balls, another with $3$ balls, and the other four bins with $5$ balls each. Let \( q' \) be the probability that every bin ends up with exactly $5$ balls. Compute the ratio \( \frac{p'}{q'} \).
**A)** 2
**B)** 3
**C)** 4
**D)** 5
**E)** 6

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