
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
Scientific research shows that in general, students' attention changes with the teacher's lecture time in a 45-minute class. Through experimental analysis, it is found that when $0\leqslant x\leqslant 8$, students' attention increases linearly. The relationship between the attention index $y$ and time $x$ (in minutes) is $y=2x+68$. After 8 minutes, the graph of the relationship between students' attention index $y$ and time $x$ takes on a parabolic shape. At the 16th minute, the students' attention index $y$ reaches its maximum value of 92, after which their attention begins to disperse until the end of the class.<br/>$(1)$ When $x=8$, the attention index $y$ is ______, and after 8 minutes, the function relationship between students' attention index $y$ and time $x$ is ______;<br/>$(2)$ If the students' attention index is not less than 80, it is considered to be in an "ideal listening state". How long does the student stay in the "ideal listening state" during a 45-minute class? (Round to the nearest minute)<br/>$(3)$ There is a key math question that the teacher must explain continuously for 24 minutes. In order to achieve better results, it is required that the minimum value of the students' attention index during these 24 minutes is maximized. At what minute after the teacher starts the class should the teacher begin explaining this question? (Round to the nearest minute) (Reference data: $\sqrt{6}≈2.449$)

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