
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
The first World Low Carbon Economy Conference was held in Nanchang, with the theme of "Energy Saving, Emission Reduction, Green Ecology". Supported by the national scientific research department, a unit undertook technical research and adopted a new process to convert carbon dioxide into a usable chemical product. It is known that the unit's monthly processing capacity is at least 400 tons and at most 600 tons. The monthly processing cost $y$ (in yuan) and the monthly processing volume $x$ (in tons) have a functional relationship that can be approximately expressed as $y=\frac{1}{2}x^2-200x+80000$.

(1) At what monthly processing volume of carbon dioxide can the unit achieve the lowest average processing cost per ton?

(2) If the profit per ton of processed carbon dioxide is 100 yuan, can the unit make a profit each month? If it can make a profit, calculate the maximum profit; if it cannot make a profit, determine the minimum subsidy from the government required for the unit to avoid losses.

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