
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
According to market analysis, for a seafood processing company in western Guangdong, when the monthly production is between 10 tons and 25 tons, the monthly total production cost $y$ (in ten thousand yuan) can be considered as a quadratic function of the monthly production $x$ (in tons). When the monthly production is 10 tons, the total monthly cost is 200,000 yuan; when the monthly production is 15 tons, the lowest monthly total cost is 175,000 yuan.  
(1) Write down the function relationship of the monthly total cost $y$ (in ten thousand yuan) with respect to the monthly production $x$ (in tons);  
(2) Given that the selling price of the product is 16,000 yuan per ton, what is the monthly production for the maximum profit;  
(3) At what monthly production is the average cost per ton the lowest, and what is the lowest cost in ten thousand yuan?

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