
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
The "Physical Health Standards for Middle School Students" stipulates that a score of $90$ or above is excellent, $80-89$ is good, $60-79$ is passing, and below $59$ is failing. In a certain school, there are 600 eighth-grade students. To understand their physical health status, 20 students were randomly selected from the eighth grade for physical health testing. Their scores were obtained and the data (scores) were organized, described, and analyzed as follows:

**Collecting Data**

| $55$ | $65$ | $71$ | $73$ | $78$ | $82$ | $85$ | $85$ | $86$ | $86$ |
|------|------|------|------|------|------|------|------|------|------|
| $86$ | $88$ | $92$ | $92$ | $93$ | $94$ | $96$ | $97$ | $99$ | $100$ |

The average score, median, and mode of the physical health test scores are $85.2bc$.

**Organizing Data**

| Level | Excellent | Good | Passing | Failing |
|-------|-----------|------|---------|---------|
| Number of Students | $a$ | $7$ | $4$ | $1$ |

| | Average Score | Median | Mode |
|---|--------------|--------|------|
| Physical Health Test Score | $85.2$ | $b$ | $c$ |

$(1)$ Write down the values of $a$, $b$, and $c$ in the table above.

$(2)$ Estimate the number of students in the eighth grade who have achieved an excellent level of physical health.

$(3)$ If a student named Xiao Huang in the eighth grade has a physical health test score of $89$, based on the information above, determine whether his score exceeds half of the students in the grade. Explain your reasoning.

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