
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]


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Let  $ n\geq 3$  be integer. Throw a  dice  $ n$  times, let  $ X_1,\ X_2,\ \cdots ,\ X_n$  be the number of the outcome in this order respectively. For  $ i \equal{} 2,\ 3,\ \cdots ,\ n$ , denote  $ A_i$  the event such that  $ X_i \equal{} X_{i \minus{} 1}$ .

(1) What's the probability  $ p_n$  that at least one of  $ A_2,\ A_3,\ \cdots ,\ A_n$  occurs?

(2) What's the probability  $ q_n$  that at least two of  $ A_2,\ A_3,\ \cdots ,\ A_n$  occurs?

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