
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 "Tuo Lake" hairy crabs have been awarded the title of "Anhui Famous Brand Agricultural Products" by the Anhui Provincial People's Government and have been rated as one of the "Top Ten Famous Crabs in China" by the China Fisheries Association. A merchant wants to understand the distribution of the quality (in grams) of 1000 Tuo Lake hairy crabs that have already been purchased. They randomly select 50 crabs from the purchased 1000 crabs, and the data obtained is shown in the table below. It is known that the real numbers $x$ and $y$ satisfy $y=2x$.

| Quality Group (g) | $[0,50)$ | $[50,100)$ | $[100,150)$ | $[150,200)$ | $[200,250)$ | $[250,300)$ | $[300,350]$ |
|-------------------|----------|------------|-------------|-------------|-------------|-------------|------------|
| Frequency         | 2        | 4          | $x$         | $y$         | 8           | 14          | 4          |

$(1)$ Fill in the blanks: $x=6$, $y=12$;  
$(2)$ The following statements are correct:  
$A$. The mode $a$ of this data set must satisfy $250\leqslant a \lt 300$  
$B$. The mean $b$ of this data set must satisfy $200\leqslant b \lt 250$  
$C$. The median $c$ of this data set must satisfy $200\leqslant c \lt 250$  
$(3)$ It is generally believed that when the weight of a crab is not less than 4 taels (i.e., 200g), the crab is considered high-quality (without considering other factors). Estimate approximately how many high-quality crabs are in this batch.

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