
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
Two people, A and B, take turns rolling a fair six-sided die, with A rolling first. The rules are as follows: if A rolls a 1, A continues to roll; otherwise, B rolls next. If B rolls a 3, B continues to roll; otherwise, A rolls next. The two follow these rules throughout the game. What is the probability $p_n$ that A will roll on the $n$th turn?

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