
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
A certain school cafeteria offers two meal options, $A$ and $B$, for lunch and dinner (each person can only choose one option each time). After statistical analysis, it was found that the probability of students choosing meal option $A$ for lunch is $\frac{2}{3}$, and the probability of choosing meal option $B$ is $\frac{1}{3}$. Given that a student chooses meal option $A$ for lunch, the probability of choosing meal option $A$ for dinner is $\frac{1}{4}$, and the probability of choosing meal option $B$ is $\frac{3}{4}$. Given that a student chooses meal option $B$ for lunch, the probability of choosing meal option $A$ for dinner is $\frac{1}{2}$, and the probability of choosing meal option $B$ is $\frac{1}{2}$.

$(1)$ If student A chooses meal option $A$ for dinner, find the probability that student A also chooses meal option $A$ for lunch.

$(2)$ Let $X$ be the number of students in a dormitory who choose meal option $B$ for dinner out of 4 students. Assuming that each student's choice of meal option is independent, find the probability distribution and the expected value of $X$.

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