
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 pocket, there are eight cards of the same size, among which three are marked with the number $1$, three are marked with the number $2$, and two are marked with the number $3$. The first time, a card is randomly drawn from the pocket and then put back. After that, a second card is drawn randomly. Let the sum of the numbers on the cards drawn the first and second times be $\xi$.

$(1)$ When is the probability of $\xi$ the greatest? Please explain your reasoning.

$(2)$ Calculate the expected value $E(\xi)$ of the random variable $\xi$.

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