
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 box $1$, there are $2$ white balls and $4$ red balls, and in box $2$, there are $5$ white balls and $3$ red balls. All balls are of the same size and shape.  
$(1)$ Without replacement, one ball is drawn from box $1$ at a time. Find the probability of drawing a red ball for the first time and then drawing another red ball for the second time.  
$(2)$ If two balls are randomly selected from box $1$ and placed into box $2$, and then one ball is randomly selected from box $2$, find the probability that the selected ball is a red ball.

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