
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 the final of a play competition for International Women's Day, there are two plays. In the first play, there are \(n\) students from 5th class A, and in the second play, there are \(n\) students from 5th class B. The audience consists of \(2n\) mothers of all \(2n\) students. The best play is chosen by voting. It is known that each mother votes for the best play with a probability of \(\frac{1}{2}\) and for the play in which her child participates with a probability of \(\frac{1}{2}\).

a) (1 point) Find the probability that the best play will win with a majority of the votes.

b) (1 point) Answer the same question if more than two classes make it to the finals.

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