
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 students, A and B, are playing a game called "Guess the Coin, Move Forward." The rules are as follows: In each round, a coin is tossed once, and both A and B guess the outcome. The guesses of A and B must be different. The player who guesses correctly moves forward $2$ steps, while the player who guesses incorrectly moves backward $1$ step. The game consists of $n$ rounds $(n\in N^{*})$. It is specified that at the beginning of the game, A and B are at the same initial position.  
$(1)$ When $n=3$, let $X$ be the difference in the number of steps between A and B at the end of the game. Find the distribution of the random variable $X$.  
$(2)$ At the end of the game, let $Y$ be the difference in the number of steps between A and B. Find $E(Y)$ and $D(Y)$ in terms of $n$.

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