
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
After the implementation of the "double reduction" policy, schools have attached importance to extended services and increased the intensity of sports activities in these services. A sports equipment store seized the opportunity and planned to purchase 300 sets of table tennis rackets and badminton rackets for sale. The number of table tennis rackets purchased does not exceed 150 sets. The purchase price and selling price are as follows:

|            | Purchase Price (元/set) | Selling Price (元/set) |
|------------|-------------------------|------------------------|
| Table Tennis Racket | $a$ | $50$ |
| Badminton Racket | $b$ | $60$ |

It is known that purchasing 2 sets of table tennis rackets and 1 set of badminton rackets costs $110$ yuan, and purchasing 4 sets of table tennis rackets and 3 sets of badminton rackets costs $260$ yuan.

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

$(2)$ Based on past sales experience, the store decided to purchase a number of table tennis rackets not less than half the number of badminton rackets. Let $x$ be the number of table tennis rackets purchased, and $y$ be the profit obtained after selling these sports equipment.

① Find the functional relationship between $y$ and $x$, and write down the range of values for $x$.

② When the store actually makes purchases, coinciding with the "Double Eleven" shopping festival, the purchase price of table tennis rackets is reduced by $a$ yuan per set $(0 < a < 10)$, while the purchase price of badminton rackets remains the same. Given that the selling price remains unchanged, and all the sports equipment can be sold out, how should the store make purchases to maximize profit?

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