
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
Two plays reached the final of a performance contest for International Women's Day. The first play involved $n$ students from 5th grade A, and the second involved $n$ students from 5th grade B. There were $2n$ mothers of all $2n$ students present at the performance. The best play is chosen by a vote from the mothers. It is known that each mother votes with a probability of $1/2$ for the best play and with a probability of $1/2$ for the play in which her child is participating.

a) Find the probability that the best play wins by a majority of votes.

b) Answer the same question if more than two classes reached the final.

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