
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
Aquatic plants require a specific type of nutrient solution. Given that each time $a (1 \leqslant a \leqslant 4$ and $a \in R)$ units of the nutrient solution are released, its concentration $y (\text{g}/\text{L})$ changes over time $x (\text{days})$ according to the function $y = af(x)$, where $f(x)=\begin{cases} \frac{4+x}{4-x} & 0\leqslant x\leqslant 2 \\ 5-x & 2\prec x\leqslant 5 \end{cases}$. If the nutrient solution is released multiple times, the concentration at a given moment is the sum of the concentrations released at the corresponding times. According to experience, the nutrient solution is effective only when its concentration is not less than $4(\text{g}/\text{L})$.

(1) If $4$ units of the nutrient solution are released only once, how many days can it be effective?

(2) If $2$ units of the nutrient solution are released first, and then $b$ units are released after $3$ days. In order to keep the nutrient solution continuously effective in the next $2$ days, find the minimum value of $b$.

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