
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
In a math club, there are $m$ boys and $n$ girls. The instructor has $m n$ cards labeled with the numbers $1, 2, 3, \ldots, m n$, each appearing exactly once. He distributes the cards to the club members, giving one card to each member until the cards run out or everyone has a card. As a result, each club member has either one card or none if there aren't enough cards. For each boy-girl pair, the sum of the numbers on their cards is calculated and written on the board. For which values of $m$ and $n$ can the numbers on the board give different remainders when divided by $m n$?

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