
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  $A$  and  $B$ .  $A$  and  $B$  alternately discard a card face up, starting with  $A$ . The game when the sum of the discards 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