
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 pharmaceutical company is conducting drug trials in animals. It is known that after each dose of $m (m > 0)$ of the drug, the concentration $y (y$ in milligrams per liter $)$ of the drug in the animal's body after $x$ hours satisfies the function $y = mf(x)$, where $f(x) = \begin{cases} -\frac{1}{2}x^{2}+2x+8, & 0 < x \leqslant 4 \\ -\frac{x}{2}-\log_{2}x+12, & 4 < x \leqslant 16 \end{cases}$. When the concentration of the drug in the animal's body is not less than $12($ in milligrams per liter $)$, it is considered effective.
(1) To ensure effectiveness within 8 hours (including 8 hours), find the minimum value of $m$;
(2) If $m = 2$, and $k$ is an integer, find the maximum value of $k$ such that the drug remains effective within $k$ hours.

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