
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
The teacher in the general affairs office of a certain middle school in Zhuhai wants to purchase chalk for teaching at the school and has a very clear way to distinguish between "high-quality products" and "non-high-quality products" in a box of chalk. A certain brand of chalk is sold by the whole box, with each box containing 20 boxes. Based on past experience, there may be some boxes of chalk that are non-high-quality products, while the rest are high-quality products. The probabilities of a box containing 0, 1, or 2 boxes of non-high-quality chalk are 0.7, 0.2, and 0.1, respectively. In order to purchase chalk from this brand, the school's general affairs teacher has designed a purchasing plan: to buy a box of chalk, randomly check 4 boxes of chalk in the box, and if there are no non-high-quality products, then purchase, otherwise do not purchase. Let event $A$ be "buying the box of chalk that was checked," and event $B_{i}$ (where $i=0,1,2$) be "there are $i$ non-high-quality products in the box."

$(1)$ Find $P(A|B_{0})$, $P(A|B_{1})$, $P(A|B_{2})$.

$(2)$ Randomly check four boxes of chalk from a box of this brand. Let $X$ be the number of non-high-quality products. Find the probability distribution of $X$.

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