
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
A chemical plant introduced an advanced production line to produce a certain chemical product. The relationship between the total production cost $y$ (in ten thousand yuan) and the annual output $x$ (in tons) can be approximately expressed as $y=\frac{x^2}{5}-48x+8000$. It is known that the maximum annual output of this production line is $210$ tons.

$(1)$ Find the annual output in tons at which the average cost per ton of product is the lowest, and calculate the lowest cost.

$(2)$ If the average factory price per ton of product is $40$ ten thousand yuan, what should the annual output be to maximize the profit? What is the maximum profit?

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