
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
The Crystal Products Factory in the East Sea had an annual output of 100,000 pieces last year, with each crystal product selling for 100 yuan and a fixed cost of 80 yuan. Starting this year, the factory invested 1 million yuan in technology costs and plans to invest an additional 1 million yuan in technology costs each year thereafter. It is expected that the output will increase by 10,000 pieces each year, and the relationship between the fixed cost $g(n)$ of each crystal product and the number of times $n$ technology costs are invested is $g(n) = \frac{80}{\sqrt{n+1}}$. If the selling price of the crystal products remains unchanged, the annual profit after the $n$th investment is $f(n)$ million yuan.  
(1) Find the expression for $f(n)$;  
(2) Starting from this year, in which year will the profit be the highest? What is the highest profit in million yuan?

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