
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
Alex needs to borrow $\$15,\!000$ from the bank. The bank offers him two options:

1. A twelve-year loan with an annual interest rate of $8\%$ compounded semi-annually. At the end of 3 years, Alex must make a payment equal to one-third of what he owes at that time. The remaining amount continues to accrue interest until the end of the twelve years, when Alex pays off the remaining balance.
   
2. A twelve-year loan with a simple annual interest rate of $10\%$, with just one lump-sum payment at the end of the twelve years.

Calculate the positive difference between the total amounts Alex has to pay back under the two schemes. Round your answer to the nearest dollar.

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