
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 that the daily sales volume $y$ (in tons) of a certain product and its selling price $x$ (in ten thousand yuan per ton, $1 < x \leqslant 5$) satisfy: when $1 < x \leqslant 3$, $y = a(x - 4)^2 + \frac{6}{x - 1}$ ($a$ is a constant); when $3 < x \leqslant 5$, $y = kx + 7$. It is known that when the selling price is 3 ten thousand yuan per ton, 4 tons of the product can be sold daily, and when the selling price is 5 ten thousand yuan per ton, 2 tons of the product can be sold daily.
(1) Find the values of $a$ and $k$, and determine the functional expression of $y$ with respect to $x$;
(2) If the sales cost of the product is 1 ten thousand yuan per ton, try to determine the value of the selling price $x$ so that the daily profit from selling the product is maximized.

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