
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 have reached the final of a competition for March 8th. In the first play, $n$ students from the 5th grade class A participated, and in the second play, $n$ students from the 5th grade class B participated. There were $2n$ mothers of all $2n$ students present at the play. The best play is chosen by the voting of the mothers. It is known that exactly half of the mothers honestly vote for the best play, while the other half always vote for the play in which their child participates.

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

b) Answer the same question if more than two plays reached the finals.

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