
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 certain natural disaster occurred in a region, causing the local tap water to be contaminated. A department decided to purify the water quality by adding a certain chemical into the water after testing. It is known that after adding a chemical with a mass of $m$ units, the concentration $y$ (mg/L) of the chemical released in the water after $x$ days satisfies $y=mf(x)$, where $f(x)=\begin{cases} & \log_{2}(x+4), & 0 < x\leqslant 4 \\ & \frac{6}{x-2}, & x > 4 \end{cases}$. The purification is considered effective when the concentration of the chemical released in the water is not less than $6$ mg/L; it is considered optimal when the concentration is not less than $6$ mg/L and not more than $18$ mg/L.

(1) If the mass of the chemical added is $m=4$, how many days in total can the tap water be effectively purified?

(2) If the mass of the chemical added is $m$, to ensure that the tap water reaches optimal purification within $7$ days (including the 7th day from the day the chemical is added), determine the range of the mass $m$ of the chemical to be added.

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