
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
Xiao Zhang is a worker in a factory. He works $8$ hours a day. It is known that he takes $170$ minutes to produce $6$ type A products and $4$ type B products, and $350$ minutes to produce $10 type A products and $10$ type B products.<br/>$(1)$ How many minutes does Xiao Zhang need to produce one type A product and one type B product, respectively?<br/>$(2)$ The daily income of the factory workers consists of a base salary of $100$ yuan and a piece-rate wage. The piece-rate wage is $a$ yuan for producing one type A product $(2 < a < 3)$ and $2.5$ yuan for producing one type B product. Xiao Zhang plans to produce a total of $28$ type A and type B products in a day. Design a production plan that maximizes his daily salary.

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