
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 a basketball shooting training organized by a certain school, each person is allowed to shoot up to 3 times. At point $A$, each successful shot earns 3 points, and at point $B$, each successful shot earns 2 points. If the sum of the scores from the first two shots exceeds 3 points, the shooting stops; otherwise, the person shoots a third time. A student has a shooting accuracy of 0.25 at point $A$ and 0.8 at point $B$. The student chooses to shoot first at point $A$ and then at point $B$ for the rest of the shots. Let $X$ represent the total score obtained by the student after the shooting training ends.  
$(1)$ Find the probability that the student shoots 3 times.  
$(2)$ Find the expected value $E(X)$ of the random variable $X$.

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