
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
This spring, regions in Southwest China such as Yunnan and Guizhou have encountered severe droughts. In the spirit of "when one place suffers, help comes from all directions," the Fengshou Agricultural Machinery Company decided to support Shangping Village by donating a total of 10 diesel generators of three different power ratings (at least one of each type) along with matching water pumps: 4 for type A, 3 for type B, and 2 for type C. Each water pump can irrigate 1 acre of farmland per hour. The requirement is for all diesel generators and their matching water pumps to operate simultaneously for one hour to irrigate 32 acres of farmland.
(1) Let the number of type A diesel generators be $x$, and the number of type B diesel generators be $y$.
   ① Express the number of type C diesel generators in terms of $x$ and $y$;
   ② Find the function relationship between $y$ and $x$;
(2) Given that the hourly costs of operating type A, B, and C diesel generators are 130 yuan, 120 yuan, and 100 yuan respectively, how should the quantities of the three types of diesel generators be arranged so that the water pumping requirements are met while minimizing the total cost $W$ of operating the diesel generators?

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