
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 an opaque bag, there are $5$ identical balls except for their colors. There are $3$ red balls labeled $A$, $B$, $C$, and $2$ black balls labeled $D$, $E$. One ball is drawn from the bag without replacement, and this process is repeated for a second time.<br/>$(1)$ Write down the sample space of this experiment;<br/>$(2)$ Let event $M$ be "drawing a red ball on the first draw" and event $N$ be "drawing a black ball on the second draw". Find the probabilities of events $M$ and $N$ occurring.

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