
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
Given that the function relationship between the monthly processing cost $y$ (in ten thousand yuan) of a certain sewage treatment plant and the monthly processing volume $x$ (in ten thousand tons) can be approximately represented as $y=\frac{1}{400}{x}^{2}-mx+25(80≤x≤210)$. When the monthly processing volume is $120$ thousand tons, the monthly processing cost is $49$ thousand yuan. The cost of processing $1$ thousand tons of sewage at this plant is $0.9$ thousand yuan.  
$(1)$ For how many thousand tons of sewage should the plant process each month to minimize the cost per thousand tons?  
$(2)$ Write down the function relationship between the monthly profit $z$ (in thousand yuan) of the plant and the monthly processing volume $x$ (in ten thousand tons), and find the maximum monthly profit.

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