
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
Around the overall goal of "building a national-level modern agricultural industry demonstration park," a county in Yunnan Province introduced various varieties of oranges with good taste to promote rural revitalization. A supermarket in the county sees the market value of two types of oranges, A and B. After investigation, it is found that the cost price of type A oranges is $a$ yuan per kilogram, and the selling price is $16$ yuan per kilogram; the cost price of type B oranges is $b$ yuan per kilogram, and the selling price is $24$ yuan per kilogram.  
$(1)$ The supermarket needs $430$ yuan to purchase 15 kilograms of type A oranges and 20 kilograms of type B oranges, and $212$ yuan to purchase 10 kilograms of type A oranges and 8 kilograms of type B oranges. Find the values of $a$ and $b$.  
$(2)$ The supermarket decides to purchase a total of $100$ kilograms of type A and type B oranges per day (both quantities are integers), with an investment of no less than $1160$ yuan and no more than $1168$ yuan. What are the possible purchasing plans for the supermarket? Which plan yields the maximum profit, and what is the maximum profit in yuan?

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