
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 very fast stain removal speed. It is known that for every $k$ ($1 \leqslant k \leqslant 4$ and $k \in R$) units of the detergent placed in a certain amount of water in a washing machine, its concentration $y$ (kg/L) in the water changes with time $x$ (minutes) according to the function $y = k \cdot f(x)$, 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 effectively remove stains when its concentration in the water is not less than $4$ (kg/L).

(1) If $k$ units of the detergent are placed, and the concentration of the detergent in the water is $4$ (kg/L) after $3$ minutes, find the value of $k$;

(2) If $4$ units of the detergent are placed, how many minutes can the effective stain removal time reach?

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