
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]

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[ACTION NAME]

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...

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system
To protect the environment and develop a low-carbon economy, a unit, with the support of the national scientific research department, has embarked on a technical breakthrough project that converts carbon dioxide into usable chemical products. The monthly processing cost $y$ (in yuan) and the monthly processing volume $x$ (in tons) of this project can be approximately represented by the function:
$$
y=\begin{cases} 
\frac{1}{3}x^{3}-80x^{2}+5040x, & x\in[120,144), \\
\frac{1}{2}x^{2}-200x+80000, & x\in[144,500),
\end{cases}
$$
and the value of the chemical products obtained from processing one ton of carbon dioxide is 200 yuan. If the project is not profitable, the state will provide compensation.

(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 from the state required to prevent the project from making a loss.

(2) At what monthly processing volume (in tons) does the project achieve the lowest average processing cost per ton?

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