
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 term "gutter oil" poses a serious threat to public health. With the support of government departments, a company has launched a new project that extracts biodiesel from "food waste". The cost of processing $y$ (in yuan) as a function of the monthly processing volume $x$ (in tons) can be approximated by: 
$$y= \begin{cases} \frac {1}{3}x^{3}-80x^{2}+5040x，& \text{for } x\in[120，144) \\ \frac {1}{2}x^{2}-200x+80000，& \text{for } x\in[144，500)\end{cases}$$
Moreover, processing one ton of "food waste" can yield biodiesel worth 200 yuan. If the project is not profitable, the government will provide subsidies.
(1) When $x\in[200，300)$, determine whether the project can be profitable. If it is profitable, calculate the maximum profit; if not, calculate the minimum monthly subsidy the government needs to provide to prevent the project from incurring losses.
(2) At what monthly processing volume does the project achieve the lowest average processing cost per ton?

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