
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
A certain company, in response to the national call for garbage classification, has launched a project with the support of the research department to innovate technologies. The project processes kitchen waste into reusable chemical products. It is known that the daily processing capacity $x$ (unit: tons) of the company is at least 70 tons and at most 100 tons. The function relationship between the total daily processing cost $y$ (unit: yuan) and the daily processing capacity $x$ can be approximately represented as $y=\frac{1}{2}x^2+40x+3200$, and the selling price of the chemical products obtained from processing 1 ton of kitchen waste is 110 yuan.  
$(1)$ For how many tons of daily processing capacity does the company have the lowest average cost per ton of kitchen waste processed? At this capacity, is the company processing 1 ton of kitchen waste at a loss or profit?  
$(2)$ In order for the company to achieve sustainable development, the government decides to provide financial subsidies to the company, with two subsidy options:  
① A fixed daily financial subsidy of 2300 yuan;  
② A financial subsidy based on the daily processing capacity, with an amount of 30x yuan. If you are the decision-maker of the company, which subsidy option would you choose to maximize profit? Why?

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