
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
In the year $2022$, a certain factory plans to produce a certain product ranging from $25$ tons to $60$ tons. It is known that the total cost $y (\text{in 10,000 yuan})$ of producing this product is related to the total output $x (\text{in tons})$ by the equation $y=\frac{x^2}{10}-2x+90$.
$(1)$ When the total output is $10$ tons, what is the total cost in thousands of yuan?
$(2)$ If the selling price of this product is $80,000 per ton, find the maximum profit the factory can make in 2022.
$(3)$ Find the minimum production cost per ton of this product.

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