
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 king called two wise men. He gave the first one 100 empty cards and ordered him to write a natural number on each card (the numbers do not have to be different), without showing them to the second wise man. The first wise man can then inform the second wise man of several different numbers, each of which is either written on one of the cards or is the sum of the numbers on some of the cards (without specifying how each number is obtained). The second wise man must determine which 100 numbers are written on the cards. If he fails, both will be beheaded; otherwise, each will lose as many hairs from their beards as the number of numbers communicated by the first wise man. How can the wise men, without discussing beforehand, stay alive and lose the minimum number of hairs?

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