
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 box contains $6$ balls, of which $3$ are yellow, $2$ are blue, and $1$ is red. Three balls are drawn from the box.

$(1)$ If the $3$ yellow balls are numbered as $A$, $B$, $C$, the $2$ blue balls are numbered as $d$, $e$, and the $1$ red ball is numbered as $x$, use $(a,b,c)$ to represent the basic event. List all the basic events of this experiment.

$(2)$ Calculate the probability of having exactly two yellow balls.

$(3)$ Calculate the probability of having at least $1$ blue ball.

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