
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
In a jewelry store display, there are 15 diamonds. Next to them are labels indicating their weights, written as $1, 2, \ldots, 15$ carats. The shopkeeper has a balance scale and four weights with masses $1, 2, 4,$ and $8$ carats. A customer is allowed only one type of weighing: place one of the diamonds on one side of the scale and the weights on the other side to verify that the weight indicated on the corresponding label is correct. However, for each weight taken, the customer must pay the shopkeeper 100 coins. If a weight is removed from the scale and not used in the next weighing, the shopkeeper keeps it. What is the minimum amount that needs to be paid to verify the weights of all the diamonds?

[8 points]

(A. V. Gribalko)

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