
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
There are 10 baskets of Chinese cabbage, with a standard weight of 20 kilograms per basket. The excess weight is recorded as a positive number, and the deficient weight is recorded as a negative number. The records are as follows:

| Difference from Standard Weight (kg) | $1$ | $2.5$ | $2$ | $-0.5$ | $1.5$ | $-1$ |
|--------------------------------------|-----|-------|-----|--------|-------|-----|
| Number of Baskets                     | $2$ | $1$   | $1$ | $2$    | $1$   | $3$ |

$(1)$ According to the records, how many kilograms heavier is the heaviest basket compared to the lightest basket among the 10 baskets of Chinese cabbage?  
$(2)$ Compared to the standard weight, how many kilograms do the 10 baskets of Chinese cabbage exceed or fall short of in total?  
$(3)$ If the purchase price of these 10 baskets of Chinese cabbage is $a$ yuan per kilogram, and the selling price is $b$ yuan per kilogram, after selling $\frac{1}{6}$ of them and selling the remaining cabbage at a 20% discount, how much profit can be made from selling these 10 baskets of cabbage? (Express your answer in terms of $a$ and $b$)

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