
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 company has invested 810,000 yuan to distribute a product over a period of 60 months. Market research shows that the company's profit function for this product during the distribution period in the $x$-th month is $f(x)=\begin{cases} & 1.1, 1\leqslant x\leqslant 20 (x\in N^*), \\\\ & \frac{1}{10}x, 21\leqslant x\leqslant 60 (x\in N^*) \\\\ \end{cases}$ (in units of 10,000 yuan). To gain more profit, the company reinvests the profit from each month into the business of the next month. Let $g(x)= \frac{\text{the profit of the }x\text{-th month}}{\text{the total capital of the }x\text{-th month}}$ denote the profit rate of the $x$-th month, for example, $g(3)=\frac{f(3)}{81+f(1)+f(2)}$.

(1) Calculate $g(10)$;
(2) Calculate the profit rate of the $x$-th month;
(3) Determine in which month the profit rate is the highest and calculate the profit for that month.

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