
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 has \( m \) gold coins and \( n \) silver coins. At the beginning of each round, the player bets some coins on red, some on black (it is allowed to place no bet on one color, and some coins can be left unbet). At the end of each round, the dealer announces that one of the colors won. The dealer returns to the player the bet on the winning color, doubling the amount of each type of coin, and takes the bet on the losing color. The player wants to have exactly three times as many coins of one type as the other (in particular, he is fine with having no money at all). For which \( m \) and \( n \) can the dealer not prevent this from happening?

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