
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 manufacturing plant has a production equipment worth ¥100,000. To enhance the production capacity and increase the added value of the product, a technical renovation is planned. After market research, it is found that the added value of the product (y, in ¥10,000) is related to the technical renovation investment (x, in ¥10,000) as follows:
(1) $y$ is positively proportional to the product of $(1-x)$ and $x^2$;
(2) When $x=0.5$, $y=0.5$, and the technical renovation investment $x$ satisfies $\frac{x}{2(1-x)}\in(0,t]$, where $t$ is a constant.
(1) Find the analytic expression of $y=f(x)$ and its domain;
(2) When $t\in(0,2]$, find the maximum added value of the product and the corresponding technical renovation investment.

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