
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
A deck of 2n cards numbered from 1 to 2n is shuffled and n cards are dealt to each player, A and B. A and B alternately discard a card face up, starting with A. The game ends when the sum of the discarded cards is first divisible by 2n + 1, and the last person to discard wins. What is the probability that A wins if neither player makes a mistake?

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