
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
As the year-end approaches, a vegetable processing point in Zhengzhou analyzed the market and found that 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, and when the monthly production is 150,000 tons, the monthly total cost is at its lowest, which 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 should the monthly production be to achieve the maximum profit, and calculate the maximum profit.

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