
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
Given the daily sales volume of a product, $y$ (in tons), in relation to its sale price, $x$ (in ten thousand yuan per ton, $1 < x \leq 5$), follows the relationship: When $1 < x \leq 3$, $y = a(x - 4)^2 + \frac{6}{x-1}$ (where $a$ is a constant); when $3 < x \leq 5$, $y = kx + 7$ (where $k < 0$). It is known that when the sale price is 3 ten thousand yuan per ton, 4 tons of the product can be sold daily, and when the sale price $x \in (3, 5]$ the lowest sales volume is 2 tons.
(1) Determine the values of $a$ and $k$, and establish the analytic expression of $y$ as a function of $x$;
(2) If the cost of the product is 1 ten thousand yuan per ton, find the value of the selling price $x$ that maximizes the daily profit from selling the product.

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