
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 invests 8.1 million yuan to distribute a product over a period of 60 months. Market research shows that the company's profit in the x-th month of distributing this product is $f(x)= \begin{cases} 1(1\leqslant x\leqslant 20,x\in N^{*}) \ \frac {1}{10}x(21\leqslant x\leqslant 60,x\in N^{*})\end{cases}$ (unit: ten thousand yuan). To gain more profit, the company invests the profit of each month into the operation of the next month. Let $g(x)= \frac {{the~profit~of~the~x-th~month}}{{the~total~funds~before~the~x-th~month}}$, for example: $g(3)= \frac {f(3)}{81+f(1)+f(2)}$.
(1) Calculate $g(10)$;
(2) Calculate the profit rate $g(x)$ of the x-th month;
(3) During the period of the company's distribution of this product, which month has the highest profit rate, and what is the profit rate of that month.

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