
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
Box A and Box B each contain $6$ balls of the same size. Box A has $2$ red balls and $4$ blue balls, while Box B has $3$ red balls and $3$ blue balls. First, $2$ balls are randomly drawn from Box A and placed into Box B, then $1$ ball is randomly drawn from Box B.  
$(1)$ The probability of drawing at least one blue ball from Box A.  
$(2)$ The probability of drawing a blue ball from Box B.

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