
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
Players \( A \) and \( B \) play a game on a blackboard that initially contains 2020 copies of the number 1. In every round, player \( A \) erases two numbers \( x \) and \( y \) from the blackboard, and then player \( B \) writes one of the numbers \( x+y \) and \( |x-y| \) on the blackboard. The game terminates as soon as, at the end of some round, one of the following holds:
1. One of the numbers on the blackboard is larger than the sum of all other numbers.
2. There are only zeros on the blackboard.

Player \( B \) must then give as many cookies to player \( A \) as there are numbers on the blackboard. Player \( A \) wants to get as many cookies as possible, whereas player \( B \) wants to give as few as possible. Determine the number of cookies that \( A \) receives if both players play optimally.

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