
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 ecological park in Hefei city responds to the national policy of developing organic agriculture and vigorously plants organic vegetables. A supermarket sees the market value of two types of organic vegetables, A and B. After investigation, it is found that the cost price of type A vegetables is $m$ yuan per kilogram, and the selling price is $16$ yuan per kilogram; the cost price of type B vegetables is $n$ yuan per kilogram, and the selling price is $18$ yuan per kilogram.  
$(1)$ The supermarket needs $430$ yuan to purchase $15$ kilograms of type A vegetables and $20$ kilograms of type B vegetables; it needs $212$ yuan to purchase $10$ kilograms of type A vegetables and $8$ kilograms of type B vegetables. Find the values of $m$ and $n$.  
$(2)$ The supermarket decides to purchase a total of $100$ kilograms of type A and type B vegetables per day, with the purchase of type A vegetables not exceeding $60$ kilograms, and the investment not exceeding $1168$ yuan. Let the purchase of type A vegetables be $x$ kilograms ($x$ is a positive integer). Find all possible purchasing plans.  
$(3)$ Under the conditions of $(2)$, when the supermarket achieves the maximum profit, it decides to donate $2a$ yuan per kilogram of type A vegetables and $a$ yuan per kilogram of type B vegetables to the local welfare institution. If the profit margin after donation is not less than $20\%$, find the maximum value of $a$.

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