
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
In a certain place where poverty alleviation efforts are being made, organic agriculture is being vigorously developed, and two types of vegetables, A and B, are being planted. A supermarket can purchase 15 kilograms of type A vegetables and 20 kilograms of type B vegetables for $430, and 10 kilograms of type A vegetables and 8 kilograms of type B vegetables for $212. <br/>$(1)$ Find the unit prices of type A and type B vegetables purchased by the supermarket. <br/>$(2)$ If the supermarket purchases a total of 100 kilograms of type A and type B vegetables each day (the weights of type A and type B vegetables are both integers), and the expenditure is not less than $1160 and not more than $1200, how many purchasing schemes does the supermarket have? <br/>$(3)$ It is known that the market selling price of type A vegetables is $16 per kilogram, and the market selling price of type B vegetables is $18 per kilogram. Under the conditions of $(2)$, the supermarket decides to purchase and sell according to the scheme that can maximize profit (all vegetables purchased each day can be sold), and the profit obtained will be donated to the local government as poverty alleviation fund at a standard of $2a per kilogram for type A vegetables and $a per kilogram for type B vegetables. 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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