
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
Maker and Breaker are building a wall. Maker has a supply of green cubical building blocks, and Breaker has a supply of red ones, all of the same size. On the ground, a row of  $m$  squares has been marked in chalk as place-holders. Maker and Breaker now take turns in placing a block either directly on one of these squares, or on top of another block already in place, in such a way that the height of each column never exceeds  $n$ . Maker places the first block.
Maker bets that he can form a green row, i.e. all  $m$  blocks at a certain height are green. Breaker bets that he can prevent Maker from achieving this. Determine all pairs  $(m,n)$  of positive integers for which Maker can make sure he wins the bet.

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