
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 certain business invested $1.44$ million US dollars to build a vegetable processing factory in a business park in mainland China. In the first year, various expenses amounted to $0.24$ million US dollars, and in the following years, the expenses increased by $0.08$ million US dollars compared to the previous year. The annual revenue from vegetable sales is $1$ million US dollars. Let $f(n)$ represent the net profit for the first $n$ years $(f(n) = $ total revenue for the first $n$ years $-$ total expenses for the first $n$ years $-$ investment amount).<br/>$(1)$ From which year onwards does the business start making a net profit?<br/>$(2)$ Five years later, the business owner decides to sell the factory to develop a new project. There are two options: ① Sell the factory for $0.96$ million US dollars when the average annual profit is maximized. ② Sell the factory for $0.32$ million US dollars when the total net profit is maximized. Which option is more profitable?

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