
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 player is playing the following game. In each turn he ﬂips a coin and guesses the outcome. If his guess is correct, he gains  $ 1$  point; otherwise he loses all his points. Initially the player has no points, and plays the game 
until he has  $ 2$  points. 
(a) Find the probability  $ p_{n}$  that the game ends after exactly  $ n$  ﬂips. 
(b) What is the expected number of ﬂips needed to ﬁnish the game?

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