
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
In the game of rolling a die and placing balls into boxes: if a 1 is rolled, a ball is placed into box A; if a 2 or 3 is rolled, a ball is placed into box B; if a 4, 5, or 6 is rolled, a ball is placed into box C. Suppose after rolling the die n times, the number of balls in boxes A, B, and C are x, y, and z, respectively.  
(1) When n=3, find the probability that x, y, and z form an arithmetic sequence;  
(2) When n=6, find the probability that x, y, and z form a geometric sequence;  
(3) Suppose after rolling the die 4 times, the absolute difference in the number of balls between box A and box B is ξ, find Eξ.

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