
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
To protect the environment and develop a low-carbon economy, a unit, with the support of the national scientific research department, has undertaken a technical challenge to launch a project that converts carbon dioxide into usable chemical products. The monthly processing cost $y$ (in yuan) and the monthly processing volume $x(t)$ of the project can be approximately represented by the function

$$
y=\begin{cases} 
\frac{1}{3}x^{3}-80x^{2}+5040x, & x\in[120,144) \\
\frac{1}{2}x^{2}-200x+80000, & x\in[144,500)
\end{cases}
$$

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 incurring losses.

$(2)$ 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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