
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
To protect the environment and develop a low-carbon economy, a unit, with the support of the national scientific research department, has tackled technical challenges and adopted a new process to convert carbon dioxide into a usable chemical product. It is known that the unit's monthly processing capacity is at least 400 tons and at most 600 tons. The monthly processing cost $y$ (in yuan) is related to the monthly processing volume $x$ (in tons) by the function: $y= \frac {1}{2}x^{2}-200x+80000$, and the value of the chemical product obtained from processing one ton of carbon dioxide is 100 yuan.
1. At what monthly processing volume, in tons, does the unit achieve the lowest average processing cost per ton?
2. Can the unit make a profit? If it can, calculate the maximum profit; if not, how much subsidy does the state need to provide at least to prevent the unit from incurring losses?

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