
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 new type of laundry detergent has a fast stain removal speed. It is known that for every $k (1 \leqslant k \leqslant 4, k \in R)$ units of detergent added to a certain amount of water in a washing machine, the concentration $y(\text{g}/\text{L})$ changes with time $x(\text{minute})$ and the function relationship is approximately $y=k \cdot f(x)$, where $f(x) = \begin{cases} (\frac{16}{9-x} - 1), & 0 \leqslant x \leqslant 5 \\ (11 - \frac{2}{45}x^{2}), & 5 < x \leqslant 16 \end{cases}.$
According to experience, the detergent can only achieve effective stain removal when its concentration in the water is no less than $4(\text{g}/\text{L}).$
(I) If $k$ units of detergent are added, and the concentration of the detergent in the water is $4(\text{g}/\text{L})$ after 3 minutes, find the value of $k.$
(II) If 4 units of detergent are added, how many minutes can the effective stain removal time reach?

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