
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
There are $n$ balls labeled $1$, $2$, $3$, ..., $n$ (all identical except for their labels) and $n$ boxes labeled $1$, $2$, $3$, ..., $n$. These $n$ balls are placed into the $n$ boxes in any way, with one ball in each box. If ball $i$ ($i=1,2,3,...,n$) is placed in box $i$, it is considered placed correctly; otherwise, it is considered placed incorrectly. Let $\xi$ represent the number of balls placed correctly.
- $(1)$ When $n=4$, find the probability of $\xi=0$.
- $(2)$ When $n=5$, find the distribution and expected value of $\xi$.

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