
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
A blind box refers to a toy box where consumers cannot know the specific product style in advance. A certain brand has launched two blind box sets. Set $A$ contains $4$ different items, including a small rabbit toy. Set $B$ contains $2$ different items, with a $50\%$ chance of getting a small rabbit toy.

$(1)$ Individuals Jia, Yi, and Bing each buy $1$ set of blind box set $B$. Let the random variable $\xi$ represent the number of small rabbit toys among them. Find the distribution table and mathematical expectation of $\xi$.

$(2)$ A consumer bought $1$ set of blind box set $A$ and $1$ set of blind box set $B$ on the first and second days of sale, respectively. After mixing all $6$ items together and randomly selecting $1$ to open, a small rabbit toy was found. Find the probability that this small rabbit toy came from blind box set $B$.

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