
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
In extracurricular sports activities, two students, A and B, play a shooting game. Each person shoots $3$ times. If they make a shot, they get $2$ points; otherwise, they get $0$ points. It is known that the probability of A making a shot each time is $\frac{1}{2}$, and the shots are independent of each other. B has a probability of $\frac{1}{2}$ of making the first shot. Starting from the second shot, if the previous shot was made, the probability of making the next shot is $\frac{3}{5}$; if the previous shot was missed, the probability of making the next shot is $\frac{2}{5}$.

$(1)$ Find the probability that A scores more than $3$ points in $3$ shots.

$(2)$ Let $X$ be the score of B in $3$ shots. Find the probability distribution and expectation of $X$.

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