
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, to repurpose waste and save resources, has initiated a project to extract biodiesel from household garbage. After calculations, it was found that the monthly processing cost $y$ (in yuan) and the monthly processing amount $x$ (in tons) can be approximated 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 usable biodiesel obtained per ton of household garbage processed is $200$ yuan. If the project does not make a profit, the government will provide subsidies.

(1) When $x \in [200, 300]$, determine whether the project can make a profit. If it is profitable, find the maximum profit; if not, calculate the minimum monthly subsidy the government needs to provide to prevent the project from incurring a loss.

(2) At what monthly processing amount does the average processing cost per ton reach its lowest?

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