
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

...

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system
The First World Low-Carbon Economy Conference was held in Nanchang. Under the support of the national scientific research department, a certain unit carried out technical tackling and adopted new processes to convert carbon dioxide into a usable chemical product. It is known that the minimum monthly processing volume of the unit is 300 tons, and the maximum is 600 tons. The relationship between the monthly processing cost $y$ (in yuan) and the monthly processing volume $x$ (in tons) can be approximately expressed as $y=\frac{1}{2}x^2-200x+45000$. Moreover, for each ton of carbon dioxide processed, a usable chemical product worth 200 yuan is obtained.
(1) How many tons should the unit process per month to minimize the average processing cost per ton?
(2) Can the unit make a profit every month? If so, find the maximum profit; if not, find the minimum subsidy the national government should provide to prevent the unit from losing money.

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