
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]


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To reduce carbon emissions, a certain company adopts a new process to convert the carbon dioxide produced during production into a chemical product. It is known that the minimum monthly processing volume of the company is 30 tons, and the maximum is 400 tons. The function relationship between the monthly processing cost $f(x)$ in yuan and the monthly processing volume $x$ (tons) is approximately represented as $f(x) = \frac{1}{2}x^2 - 300x + 64800$.

$(1)$ At what monthly processing volume does the company achieve the lowest monthly processing cost? What is the lowest monthly processing cost in yuan?

$(2)$ At what monthly processing volume does the company achieve the lowest average processing cost per ton? What is the lowest average processing cost per ton in yuan?

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