
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
Manuel is a mathematician who likes card games. During a bus trip, he encounters the brothers Jonas and Jonatan and proposes a game. Only the four aces from the deck will be used: the hearts and diamonds are red, while the spades and clubs are black.

Manuel will be the dealer, and the two brothers will each bet 1 real against him in each round, taking turns. The cards are placed face down. Jonas picks a card, and Jonatan turns it face up. Jonas then picks another card, and Jonatan again turns it face up. If both cards are the same color, Jonas wins 1 real from Manuel. Otherwise, Manuel wins 1 real from Jonas. Then, Jonas and Jonatan switch roles, and the game continues. Manuel does not touch the cards and thus cannot manipulate the game. Jonatan thinks for a bit and concludes he has a \(\frac{2}{3}\) chance of winning, since the outcomes are only two red cards, two black cards, or one red and one black. Is this correct?

(a) Jonas has participated in mathematics olympiads and decided to be more cautious. He decided to analyze this game using a probability tree. What would Jonas's probability tree look like?

(b) Considering the results of the tree from the previous part, what is the probability that Manuel wins each round of the game?

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