
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
We wish to distribute 12 indistinguishable stones among 4 distinguishable boxes \( B_{1}, B_{2}, B_{3}, B_{4} \). (It is permitted some boxes are empty.)

(a) Over all ways to distribute the stones, what fraction of them have the property that the number of stones in every box is even?

(b) Over all ways to distribute the stones, what fraction of them have the property that the number of stones in every box is odd?

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