
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 order to protect the environment, a factory, following the national call, recycles waste into a certain product. It has been calculated that the relationship between the processing cost $y$ (in ten thousand yuan) and the processing amount $x$ (in tons) can be approximately represented by: $y=x^2-50x+900$, and for every ton of waste processed, a product worth ten thousand yuan can be obtained, along with a national subsidy of ten thousand yuan.  
(1) When $x \in [10, 15]$, determine whether this initiative can be profitable. If it can be profitable, calculate the maximum profit; if it cannot be profitable, calculate the minimum amount of national subsidy in ten thousand yuan required for the factory not to incur losses.  
(2) At what processing amount is the average processing cost per ton the lowest?

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