
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 car manufacturing company has a production line worth $a$ million dollars. The company aims to increase the production capacity of this line and enhance the product's added value $y$ through technical renovation with an investment of $x$ million dollars. After market research, it was found that the product's added value $y$ and the technical renovation investment $x$ satisfy the following conditions: (1) $y$ is directly proportional to the product of $(a-x)$ and $x^2$; (2) $x \in (0, \frac{2am}{2m+1}]$, where $m$ is a constant. If $x = \frac{a}{2}$, then $y = a^3$.

(1) Find the expression for the product's added value $y$ in terms of $x$.
(2) Find the maximum value of the product's added value $y$ and the corresponding value of $x$.

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