
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]


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Guifei mango, also known as Red Jinlong, is a type of fruit produced in Hainan. The mango can be classified into four grades: Grade $A$, Grade $B$, Grade $C$, and Grade $D$. A certain purchaser plans to order a batch of these mangoes to sell outside the province. They randomly select $100$ boxes from the purchased batch of mangoes and obtain the following data based on the classification standard of the mangoes (treating sample frequency as probability):

| Grade | $A$ | $B$ | $C$ | $D$ |
|-------|-----|-----|-----|-----|
| Number of Boxes | $40$ | $30$ | $20$ | $10$ |

$(1)$ From these $100$ boxes of mangoes, $4$ boxes are randomly selected with replacement. Let $\xi$ be the number of Grade $A$ boxes among these $4$ boxes. Find the probability $P\left(\xi = 2\right)$ and the mathematical expectation of $\xi$.

$(2)$ Using the sample to estimate the population, the orchard owner proposes two options for the purchaser to consider.

Option 1: Sell without classification at a price of $30$ yuan per box.

Option 2: Sell by grade, with mango prices as shown in the table below.

| Grade | $A$ | $B$ | $C$ | $D$ |
|-------|-----|-----|-----|-----|
| Price ($/box) | $38$ | $32$ | $26$ | $16$ |

From the perspective of the purchaser, which option should be chosen?

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