
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
There are several black and white pieces placed on a circumference. Two players, A and B, play a game: Player A takes all black pieces that are adjacent to any white piece, then Player B takes all white pieces that are adjacent to any black piece. They take turns in this manner until only pieces of one color remain.
1. Suppose there are initially 40 pieces on the circumference. Is it possible to have only one piece remaining after each player has taken pieces twice?
2. Suppose there are initially 1000 pieces on the circumference. What is the minimum number of turns needed to have only one piece remaining on the circumference?

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