
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
Two people play the following game. They have a chocolate bar divided by grooves, parallel to the sides of the bar, into pieces. The grooves break the bar into M vertical and N horizontal strips. The first player on their turn breaks the bar along one of the grooves into two rectangular parts and gives them to the second player. The second player chooses one of the parts, eats it, and breaks the other part along a groove into two parts and gives them to the first player. The first player eats one of the parts, breaks the other, and gives it back, and the process repeats. The player who cannot make a move loses. For which M and N can the first player play in such a way that they win regardless of the second player's actions?

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