
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
At the beginning of the school year, a certain school conducted a mathematics quiz (out of $100$ points) to understand the mathematics learning situation of eighth-grade students during the winter vacation. They randomly selected $10$ students from each of Class A and Class B for statistical analysis. The collected data is as follows:

Class A: $90$, $90$, $70$, $90$, $100$, $80$, $80$, $90$, $95$, $65$.

Class B: $95$, $70$, $80$, $90$, $70$, $80$, $95$, $80$, $100$, $90$.

Organize the data:

| Score $x$ | $60\leqslant x\leqslant 70$ | $70 \lt x\leqslant 80$ | $80 \lt x\leqslant 90$ | $90 \lt x\leqslant 100$ |
|-----------|---------------------------|-----------------------|-----------------------|------------------------|
| Class A   | $2$                       | $2$                   | $4$                   | $2$                    |
| Class B   | $2$                       | $3$                   | $a$                   | $3$                    |

Analyze the data:

| Data    | Mean  | Median | Mode |
|---------|-------|--------|------|
| Class A | $85$  | $90$   | $d$  |
| Class B | $b$   | $c$    | $80$ |

Answer the following questions:

$(1)$ Directly write down the values of $a$, $b$, $c$, and $d$.

$(2)$ Xiao Ming said, "I scored $90$ points on this quiz, which is considered above average in our group!" Based on the table above, determine which class Xiao Ming might belong to and explain your reasoning.

$(3)$ If there were a total of $50$ students from Class B taking the quiz, estimate the number of students from Class B who scored above $90$ points.

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