
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
Thirty-six players are playing a game: they take turns drawing a random card from a 36-card deck. If a player draws the ace of diamonds, they win. If they draw any other card, they return it to the deck, and the next player takes their turn. This continues in a cycle: first player, then the second, and so on. If the ace of diamonds is not drawn in the first round, the players continue drawing cards in the same order in the second round, and so on. This process repeats until someone draws the ace of diamonds.

Assume that before the game starts, the players place bets, and the winner takes all. How should the players' bets be related so that the game is fair, meaning the expected winnings for all players are equal to each other (i.e., equal to zero taking into account the amount each player bet)?

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