
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
Consider a three-person game involving the following three types of fair six-sided dice.
- Dice of type $A$ have faces labelled 2, 2, 4, 4, 9, 9.
- Dice of type $B$ have faces labelled $1, 1, 6, 6, 8, 8$.
- Dice of type $C$ have faces labelled $3, 3, 5, 5, 7, 7$.
All three players simultaneously choose a die (more than one person can choose the same type of die, and the players don't know one another's choices) and roll it. Then the score of a player $P$ is the number of players whose roll is less than $P$'s roll (and hence is either 0, 1, or 2). Assuming all three players play optimally, what is the expected score of a particular player?

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