
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
$(1)$ Given $x \gt 1$, $y \gt 1$, and $x+y=4$. Find the minimum value of $\frac{1}{x-1}+\frac{1}{y-1}$. <br/>$(2)$ A unit, supported by the national scientific research department, is conducting technological research and development. By using a new process, carbon dioxide is converted into a usable chemical product. It is known that the unit's monthly processing volume ranges from a minimum of 400 tons to a maximum of 600 tons. The relationship between the monthly processing cost $y$ (in yuan) and the monthly processing volume $x$ (in tons) can be approximately represented by $y=\frac{1}{2}{x}^{2}-200x+80000$. For what monthly processing volume does the unit achieve the lowest average processing cost per ton, and what is this minimum cost?

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