
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 geological disaster has caused the local tap water to become contaminated in a certain area. To purify the water, a department decided to add a medication into the water. It is known that after adding a medication of mass $m$, the concentration $y$ (milligrams per liter) of the medication in the water after $x$ days satisfies $y = mf(x)$, where $$f(x) = \begin{cases} \frac{x^2}{16} + 2 & (0 < x \leq 4) \\ \frac{x + 14}{2x - 2} & (x > 4) \end{cases}$$
When the concentration of the medication in the water is not less than 4 milligrams per liter, it is considered effective purification; when the concentration of the medication in the water is not less than 4 milligrams per liter and not more than 10 milligrams per liter, it is considered optimal purification.

1. If the mass of the added medication is $m = 4$, how many days can the tap water maintain effective purification?
2. If the mass of the added medication is $m$, in order to achieve optimal purification of the tap water within 7 days (including the 7th day) after adding the medication, determine the minimum value of the mass $m$ of the medication to be added.

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