
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 distribution store purchases two types of seasonal fruits, A and B, from a farm every day for sale. The cost price and selling price of the two fruits are as follows:

| Type | Cost Price (元/kg) | Selling Price (元/kg) |
|------|---------------------|-----------------------|
| A    | $a$                 | $5$                   |
| B    | $b$                 | $7$                   |

The cost price of fruit B is $2.5$ yuan/kg higher than that of fruit A. When the fruit distribution store purchases $400$ kg of fruit B and $200$ kg of fruit A, the ratio of the cost of purchasing fruit B to the cost of purchasing fruit A is $24:7$.

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

$(2)$ The fruit distribution store purchases a total of $300$ kg of the two fruits every day and sells them all on the same day. The store must sell at least $80$ kg but no more than $120$ kg of fruit A each day. Let $x$ represent the amount of fruit A sold each day. The total profit $W$ from selling these two fruits on that day (without considering losses during the sales process) is:

① Find the relationship between $W$ and $x$, and determine the maximum profit from selling these two fruits on that day.

② On weekends, the fruit distribution store offers discounts on fruit A by reducing the selling price by $m$ yuan/kg. In order to ensure that the minimum value of the total profit from selling these two fruits on that day is not less than $312$ yuan, find the maximum value of $m$.

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