
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
Xiaoyu, Xiaohua, and Xiaoxing participated in a social practice activity at a supermarket during the summer vacation. They were involved in the sales of a certain fruit, knowing that the cost price of the fruit is $8$ yuan per kilogram. Through market research, they found that when the selling price is $10$ yuan, they can sell $300$ kilograms per day. For every increase of $1$ yuan in the selling price, the daily sales volume decreases by $50$ kilograms.  
$(1)$ Find the functional relationship between the daily sales volume $y$ (in kilograms) of this fruit at the supermarket and the selling price $x$ (in yuan per kilogram).  
$(2)$ After a period of time, it was found that the daily sales volume of this fruit is always not less than $250$ kilograms. What is the maximum profit $w$ (in yuan) that the supermarket can make from selling this fruit per day at this point?  
$(3)$ In response to the government's call, the supermarket decided to donate $a$ yuan of profit (where $a \leq 2.5$) to the Hope Project for every kilogram of this fruit sold during the summer vacation. The company found through sales records that when the selling price does not exceed $13$ yuan, the daily net profit after donation increases with the increase in the selling price $x$ (in yuan per kilogram). Find the range of values for $a$.

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