
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 certain type of orange, known for its sweetness and thin skin, is a variety of sweet oranges. This orange can be classified into four grades: premium, special, superior, and first grade. A purchaser plans to order a batch of oranges to be sold outside the province, and randomly selects 100 boxes (each containing 5kg) from this batch of oranges. The data obtained based on the classification standard of oranges is as follows:

| Grade | Premium | Special | Superior | First Grade |
|-------|---------|---------|----------|-------------|
| Number of Boxes | 40 | 30 | 10 | 20 |

$(1)$ From these 100 boxes of oranges, if one box is randomly selected, find the probability that the box is of premium grade.  
$(2)$ 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 $27$ yuan/kg;  
Option 2: Sell with classification, with prices as shown in the table below:

| Grade | Premium | Special | Superior | First Grade |
|-------|---------|---------|----------|-------------|
| Price (yuan/kg) | $36$ | $30$ | $24$ | $18$ |

From the purchaser's perspective, which option should be chosen?  
$(3)$ From the 100 boxes, if 3 boxes are randomly selected and the probability that these 3 boxes have different grades is denoted as $p_{1}$; using stratified random sampling, 20 boxes are selected from the 100 boxes, and then 3 boxes are randomly selected from the 20 boxes, with the probability that these 3 boxes have different grades denoted as $p_{2}$. Write down the relationship between $p_{1}$ and $p_{2}$ directly (no need to explain the reason).

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