
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 and some on black (it is possible to not bet on one of the colors at all, and some coins can be left out of the bet). At the end of each round, the dealer announces that one of the colors has won. The dealer returns the bet on the winning color to the player, doubling the amount of each type of coin in the bet, and keeps the bet on the losing color. The player wants to have exactly three times as many coins of one type as the other (including the possibility of having no money left 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}
user