
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 made it to the final of a competition held on March 8th. The first play featured $n$ students from 5th grade A, and the second play featured $n$ students from 5th grade B. The audience consisted of $2n$ mothers of all $2n$ students. The best play is selected by a vote from the mothers. It is known that exactly half of the mothers vote honestly for the best play, while the other half will vote for the play that their child is in, regardless of its quality.

a) Find the probability that the best play will win by a majority vote.

b) Answer the same question if more than two plays made it to the final.

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