
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 A, there are 5 red balls, 2 white balls, and 3 black balls. In box B, there are 4 red balls, 3 white balls, and 3 black balls. First, randomly select 1 ball from box A and put it into box B. Let $A_{1}$, $A_{2}$, and $A_{3}$ represent the events of selecting a red ball, white ball, and black ball from box A, respectively. Then, randomly select 1 ball from box B. Let $B$ represent the event of selecting a red ball from box B. Find $P(B|A_{1})=$____ and $P\left(B\right)=$____.

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