
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
Under the influence of the policy of creating "economic brand characteristic brand" in Xiangyang City, every evening, a popular barbecue restaurant in the city is crowded with diners. The seafood skewers and meat skewers of this barbecue restaurant are very popular. The owner purchases and processes the two products in bulk from a food processing factory for sale. The cost of seafood skewers is $m$ yuan per skewer, and the cost of meat skewers is $n$ yuan per skewer. The quantities and costs of purchasing and processing seafood skewers and meat skewers for two times are shown in the table below (cost includes purchase price and other expenses):

| Times | Quantity of Seafood Skewers | Quantity of Meat Skewers | Total Cost (yuan) |
|-------|-----------------------------|-------------------------|-------------------|
| First time | 3000 | 4000 | 17000 |
| Second time | 4000 | 3000 | 18000 |

For group consumption, the owner decides that when the consumption of seafood skewers does not exceed 200 skewers each time, the selling price is 5 yuan per skewer; for the part exceeding 200 skewers, it is discounted by 20%. The selling price of each meat skewer is 3.5 yuan.

$(1)$ Find the values of $m$ and $n$;  
$(2)$ On May 1st, a tourist group went to this restaurant to eat barbecue. They consumed a total of 1000 skewers of seafood and meat, with the seafood skewers not exceeding 400 skewers. In this consumption, let the group consume $x$ seafood skewers, and the owner's total profit from seafood skewers is $y$ yuan. Find the functional relationship between $y$ and $x$, and write down the range of values for the independent variable $x$;  
$(3)$ Under the conditions of $(2)$, if the tourist group consumes more than 200 seafood skewers, the owner decides to offer more discounts to the group by reducing the price of each meat skewer by $a$ yuan $(0 < a < 1)$, but ensuring that the total profit from meat skewers in this consumption is always not less than the total profit from seafood skewers. Find the maximum value of $a$.

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