
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
Players A and B each take two dice with uniform density to play a game, with the following rules: if the sum of the points rolled is a multiple of $3$, the original roller continues to roll; if the sum of the points rolled is not a multiple of $3$, the turn passes to the other player. It is stipulated that player A rolls first.

$(1)$ Find the probability that player A will roll in the second round.

$(2)$ Find the probability that player B will roll exactly $2$ times in the first $4$ rounds.

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