
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 play a dice throwing game, taking turns to throw a fair dice. The game ends when one player rolls a $6$. <br/>$(1)$ Let $X$ be the total number of dice throws by both players. If each player can throw the dice at most twice, find the probability distribution and expectation of $X$ when the game ends; <br/>$(2)$ Given that player A starts, find the probability that player A throws the dice exactly $n$ times and wins.

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