
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 company, aiming to turn waste into treasure and save resources, launched a project to extract biodiesel from household garbage. The monthly processing cost $y$ (in yuan) and the monthly processing volume $x$ (in tons) of the project can be approximately represented by the function: 
$$
y= \begin{cases} 
\frac {1}{3}x^{3}-80x^{2}+5040x, & x\in [120,144) \\
\frac {1}{2}x^{2}-200x+80000, & x\in [144,500)
\end{cases}
$$
Moreover, processing one ton of household garbage can generate a value of 200 yuan from the usable biodiesel. If the project is not profitable, the government will provide subsidies.
- (Ⅰ) 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 losing money.
- (Ⅱ) At what monthly processing volume $x$ can the project achieve the lowest average processing cost per ton?

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