
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
A fruit company purchased 2000 boxes of lychees at a cost of $10/kg$. Each box weighs 5 kg. Before selling the lychees, the company needs to remove the damaged ones. Twenty boxes were randomly selected, and the weights of the boxes after removing the damaged lychees are as follows:  
$4.7, 4.8, 4.6, 4.5, 4.8, 4.9, 4.8, 4.7, 4.8, 4.7$  
$4.8, 4.9, 4.7, 4.8, 4.5, 4.7, 4.7, 4.9, 4.7, 5.0$  
Organizing the data:  

| Weight (kg) | 4.5 | 4.6 | 4.7 | 4.8 | 4.9 | 5.0 |
|-------------|-----|-----|-----|-----|-----|-----|
| Quantity    | 2   | 1   | 7   | $a$ | 3   | 1   |

Analyzing the data:  

| Mean   | Mode ($b$) | Median ($c$) |
|--------|------------|--------------|
| 4.75   | $b$        | $c$          |

$(1)$ Write down the values of $a$, $b$, and $c$ in the table above.  
$(2)$ The mean, mode, and median can all reflect the central tendency of this data set. Based on the sample data analysis results, choose any one of these statistical measures and estimate how many kilograms of lychees were damaged in total out of the 2000 boxes.  
$(3)$ Based on the result in $(2)$, calculate how much the company should sell each kilogram of lychees for to break even (round to one decimal place).

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