
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 summoned two wise men. He gave the first one 100 blank cards and instructed him to write a positive number on each (the numbers do not have to be different), without showing them to the second wise man. Then, the first wise man can communicate several distinct numbers to the second wise man, each of which is either written on one of the cards or is a sum of the numbers on some cards (without specifying exactly how each number is derived). The second wise man must determine which 100 numbers are written on the cards. If he cannot do this, both will be executed; otherwise, a number of hairs will be plucked from each of their beards equal to the amount of numbers the first wise man communicated. How can the wise men, without colluding, stay alive and lose the minimum number of hairs?

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