
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 king summons two wizards to the palace. He asks Wizard A to write 100 positive real numbers on a card (duplicates allowed) and keep it secret from Wizard B. Wizard B must then accurately write down all 100 positive real numbers; if he fails, both wizards will be executed. Wizard A is allowed to provide a series of numbers to Wizard B, where each number is either one of the 100 positive real numbers or a sum of several of those numbers. However, Wizard A cannot indicate which numbers are the individual real numbers and which are the sums. The king decides to pull an equal number of beard hairs from both wizards according to the number of numbers in the series. Without the wizards being able to communicate beforehand, how many beard hairs will the wizards need to lose at minimum to avoid execution?

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