
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
Roll a die and observe the number of points facing up. Let $A=$"the number is $1$," $B=$"the number is even," $C=$"the number is less than $3$," $D=$"the number is greater than $2$," $E=$"the number is a multiple of $3$."<br/>$(1)$ Write the sample space and the events above in set notation;<br/>$(2)$ What are the relationships between events $A$ and $C$, $C$ and $D$, $D$ and $E$?<br/>$(3)$ Represent the events $\overline{D}$, $\overline{A}C$, $\overline{B} \cup C$, $\overline{D} + \overline{E}$ in set notation.

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