
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 and $n$ silver coins. At the beginning of each round, the player bets some coins on red and some on black (they can abstain from betting on one of the colors, or even place no bets at all). At the end of each round, the dealer announces which color has won. The bet on the winning color is returned to the player, doubling the number of coins of each type in the bet, while the bet on the losing color is taken by the dealer. The player wants to have exactly three times more coins of one type than the other (it is also acceptable for the player to end up with no money at all). For which values of $m$ and $n$ can the dealer not prevent the player from achieving this goal?

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