
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
The 20 pupils in a class each send 10 cards to 10 (different) class members. <span style="font-size:92%">*[note: you cannot send a card to yourself.]*</span>
(a) Show at least 2 pupils sent each other a card.
(b) Now suppose we had  $n$  pupils sending  $m$  cards each. For which  $(m,n)$  is the above true? (That is, find minimal  $m(n)$  or maximal  $n(m)$ )

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