
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 fruit planting base responded to the government's call to vigorously plant high-quality fruits. A supermarket sees the market value of two high-quality fruits, A and B. After investigation, the cost price and selling price of these two fruits are as shown in the table below:

| Fruit Type | Cost Price (元/$/kg) | Selling Price (元/$/kg) |
|------------|-----------------------|-------------------------|
| A          | $a$                   | $20$                    |
| B          | $b$                   | $23$                    |

The supermarket needs $305$ yuan to purchase $15$ kg of fruit A and $5$ kg of fruit B, and $470$ yuan to purchase $20$ kg of fruit A and $10$ kg of fruit B.

$(1)$ Find the values of $a$ and $b$.

$(2)$ The supermarket decides to purchase a total of $100$ kg of fruits A and B for sale each day, with the quantity of fruit A not less than $30$ kg and not more than $80$ kg. During actual sales, if the quantity of fruit A exceeds $60$ kg, the excess will be sold at a discount of $3$ yuan per kg. Find the relationship between the profit $y$ (in yuan) obtained by the supermarket when selling these two fruits and the quantity $x$ (in kg) of fruit A purchased, and write down the range of values for $x$.

$(3)$ Under the conditions of $(2)$, when the profit $y$ obtained by the supermarket is maximized, the supermarket decides to reduce the price of fruit A by $3m$ yuan per kg and the price of fruit B by $m$ yuan per kg. If the profit margin (profit margin $= \frac{profit}{cost}$) is not less than $16\%$, find the maximum value of $m$.

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