
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 store purchased a batch of apples at $60$ yuan per box, totaling $400$ boxes. To calculate the total weight, $30$ boxes of apples were randomly selected and weighed. It was found that each box of apples weighed around $10$ kilograms. Using $10$ kilograms as the standard weight, numbers exceeding $10$ kilograms were recorded as positive, and numbers less than $10$ kilograms were recorded as negative. The weighing records are as follows:

| Weight Difference | $-0.2$ | $-0.1$ | $0$ | $0.1$ | $0.2$ | $0.5$ |
|-------------------|--------|--------|-----|-------|-------|-------|
| Number of Boxes   | $5$    | $8$    | $2$ | $6$   | $8$   | $1$   |

$(1)$ Find the total weight of the 30 boxes of apples.  
$(2)$ If the selling price of apples per kilogram is $10$ yuan, how much profit is made from selling this batch of apples?

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