
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
A and B take turns rolling a die, with A going first. The rules are: if A rolls a 1, A continues to roll; otherwise, it's B's turn. If B rolls a 3, B continues to roll; otherwise, it's A's turn. They keep playing by these rules. Find the probability $p_{n}$ that the $n$-th roll is by A.

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