
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
In order to meet the needs of the community residents for physical fitness, the government of Guangrao County is preparing to purchase a certain number of sets of fitness equipment to provide to the community for free. After investigation and understanding, Guangyue Company has two models, $A$ and $B$, available for selection. It is known that in 2021, the price of each set of type $A$ fitness equipment from Guangyue Company is $25,000 yuan, and the price of each set of type $B$ fitness equipment is $20,000 yuan. In 2023, the price of each set of type $A$ fitness equipment is $16,000 yuan, and the annual average decline rate of both type $A$ and type $B$ fitness equipment is the same.
$(1)$ Find the annual average decline rate of each set of type $A$ fitness equipment from 2021 to 2023;
$(2)$ In 2023, the government decided to purchase and install a total of 80 sets of type $A$ and type $B$ fitness equipment from Guangyue Company through bidding. The total special funds for government procurement do not exceed 112,000 yuan, and the cost of purchasing type $A$ equipment cannot exceed the cost of purchasing type $B$ equipment. Determine how many sets of type $B$ fitness equipment should be purchased at least.

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