
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
In the winter of 2022, the new coronavirus pandemic resurged in China, affecting many cities and regions with the Omicron variant of the virus. In order to control the epidemic, a certain unit purchased a new type of air disinfectant for environmental disinfection. It is known that within a certain range, the concentration $y$ (unit: milligrams per cubic meter) of the disinfectant released into the air varies with time $x$ (unit: hours) as follows: when $0\leqslant x\leqslant 4$, $y=\frac{{16}}{{8-x}}-1$; when $4 \lt x\leqslant 10$, $y=5-\frac{1}{2}x$. If sprayed multiple times, the concentration of the disinfectant in the air at a certain moment is the sum of the concentrations released at the corresponding times. From experiments, it is known that the disinfectant concentration in the air must not be less than $4$ (milligrams per cubic meter) in order to effectively kill the viruses in the air.

$(1)$ If $4$ units of disinfectant are sprayed at once, how many hours can the effective killing time last?

$(2)$ If $2$ units of disinfectant are sprayed for the first time, and $a$ units of disinfectant are sprayed $6$ hours later, where $a$ is between $1$ and $4$, in order to maintain effective disinfection for the next $4$ hours, find the minimum value of $a$. (Round to $0.1$, reference data: $\sqrt{2}$ is approximately $1.4$)

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