Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: Luke is borrowing $\$10{,}000$ from the bank. The bank offers him a choice between two $10$-year payment plans:

${\bf Plan~1.}$  Luke's debt accumulates $10\%$ annual interest which compounds quarterly. Luke pays off half his balance after $5$ years, and the rest at the end of the $10$ years.

${\bf Plan~2.}$  Luke's debt accumulates $10\%$ annual interest which compounds annually. Luke pays off his full balance at the end of the $10$ years.

What is the (positive) difference between Luke's total payments under Plan 1 and his total payments under Plan 2? Round to the nearest dollar.
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