
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
Play a game of rolling dice and placing balls into boxes: if rolling 1, put one ball into Box A; if rolling 2 or 3, put one ball into Box B; if rolling 4 or 5 or 6, put one ball into Box C. Assuming after n rolls, the number of balls in Box A, Box B, and Box C is x, y, z respectively:
(1) When n=3, find the probability that x, y, z form an arithmetic sequence;
(2) When n=6, find the probability that x, y, z form a geometric sequence;
(3) Assuming after rolling 4 times, the absolute difference in the number of balls between Box A and Box B is ξ, calculate Eξ (the expected value of ξ).

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