
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 two opaque bags, bag A contains 2 red balls and 3 white balls, bag B contains 3 red balls and 2 white balls, and all balls are of the same size and texture.  
$(1)$ From these 10 balls, 4 balls are randomly selected. Let event A be "exactly 2 red balls are drawn from the 4 balls, and these 2 red balls come from the same bag." Find the probability of event A occurring.  
$(2)$ First, 2 balls are randomly selected from bag A and placed into bag B. Then, 2 balls are randomly selected from bag B. Find the probability that both balls selected from bag B are red.

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