
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 are playing a basketball shooting game, taking turns in each round. In each round, A and B each take one shot. The rules are as follows: if A makes the shot and B misses, A gets 1 point and B gets -1 point; if A misses and B makes the shot, A gets -1 point and B gets 1 point; if both A and B make or miss the shot, both get 0 points. The game stops when the difference in cumulative points between A and B is greater than or equal to 4 points. The player with more points wins. If after 4 rounds the difference in cumulative points between A and B is less than 4 points, the game stops and the player with more points wins. If the points are equal, it is a tie. The shooting accuracy of A and B are 0.5 and 0.6 respectively, and do not affect each other. In a round of the game, A's score is denoted as $X$.
- $(1)$ Find the distribution of $X$;
- $(2)$ Find the probability of A and B ending in a tie;
- $(3)$ Let the total number of rounds played by A and B be denoted as $Y$, find the distribution and expectation of $Y$.

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