
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
John, Emma, and Lucas play a game where John starts with $3, Emma starts with $2, and Lucas starts with $1. A bell rings every $10 seconds, at which time each of the players who currently have money independently chooses one of the other players still in the game at random and gives $1 to that player. If a player's amount reaches $0, they exit the game and can no longer participate or receive money. What is the probability that after the bell has rung $1002$ times, each remaining player will have $2?
**A)** $\frac{1}{9}$
**B)** $\frac{1}{18}$
**C)** $\frac{1}{27}$
**D)** $\frac{1}{36}$
**E)** $\frac{1}{45}$

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