
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
Hydroponic plants require a special nutrient solution. It is known that for every $a$ (where $1 \leqslant a \leqslant 4$ and $a \in \mathbb{R}$) units of nutrient solution added, the concentration $y$ (grams per liter) released in the water as a function of time $x$ (days) is approximately $y=af(x)$, where $f(x)= \begin{cases} \dfrac {4+x}{4-x} & (0 \leqslant x \leqslant 2) \\ 5-x & (2 < x \leqslant 5) \end{cases}$. If the nutrient solution is added multiple times, the concentration of the nutrient solution in the water at any given time is the sum of the concentrations released by each addition at that time. According to experience, the nutrient solution is effective only when its concentration in the water is not less than $4$ grams per liter.  
$(1)$ If only $4$ units of nutrient solution are added once, how many days can the effective time possibly reach?  
$(2)$ If $2$ units of nutrient solution are added first, and $b$ units are added $3$ days later. To ensure that the nutrient solution remains effective for the next $2$ days, find the minimum value of $b$.

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