
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
Can $4$ red balls and $3$ white balls in can A, and $3$ red balls and $2$ white balls in can B (balls are the same in size and texture except for color).<br/>$(1)$ If $3$ balls are drawn from can A, let $\xi$ be the number of red balls drawn, find the distribution and expectation of $\xi$.<br/>$(2)$ If a ball is randomly drawn from can A and placed into can B, $A_{1}$, $A_{2}$ represent whether the ball drawn from can A is red or white; then a ball is randomly drawn from can B, $B$ represents whether the ball drawn from can B is red. Find $P(B|A_{1})$ and $P(B)$.

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