
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