
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
Steaks are mainly divided into filet steak, ribeye steak, sirloin steak, and T-bone steak. A beef purchaser randomly selects 100 boxes from a batch of steaks and obtains the following data based on the classification standard of steaks:

| Type of Steak | Filet Steak | Ribeye Steak | Sirloin Steak | T-bone Steak |
|---------------|-------------|--------------|---------------|-------------|
| Quantity/Box  | 20          | 30           | 20            | 30          |

$(1)$ Using the proportional allocation stratified random sampling method, 10 boxes of steaks are selected from these 100 boxes. Then, 4 boxes are randomly selected from the 10 boxes. Find the probability that exactly 2 boxes of steaks are T-bone steaks.

$(2)$ If the frequency is considered as probability and used to estimate the population, 3 boxes are randomly selected from this batch of steaks. Let $X$ represent the number of filet steaks drawn. Find the distribution table and the expected value of $X$.

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