
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 produces two types of products, $A$ and $B$. According to market research and predictions, the monthly profit $y=f(x)$ of product $A$ is directly proportional to the investment $x$, and when the investment is 40,000 yuan, the monthly profit is 20,000 yuan. The monthly profit $y=g(x)$ of product $B$ is directly proportional to the arithmetic square root of the investment $x$, and when the investment is 40,000 yuan, the monthly profit is 10,000 yuan. (It is allowed to invest in only one type of product)

1. Find the functional relationship between the monthly profit of products $A$ and $B$ and the investment amount $x$, respectively.
2. The company has raised 100,000 yuan in funds and will invest all of it in the production of products $A$ and $B$. How should these 100,000 yuan in funds be allocated to maximize the company's monthly profit? What is the maximum monthly profit? (Express the result as a fraction)

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