Claire has borrowed $\$5,\!000$. She plans to pay off the loan in full after two payments. She will make one payment in 3 years, then another payment in 6 years. The second payment will be exactly double the amount of the first payment. How much is the first payment if the interest rate of the loan is $8.5\%$, compounded annually? Express your answer as a dollar value rounded to the nearest cent.

To find the amount of the first payment, we need to consider the present value of the loan. The present value is the amount of money needed to pay off a loan in the future, based on the interest rate and time periods.

In this case, Claire will make two payments, one in 3 years and another in 6 years. The second payment will be double the amount of the first payment. So, let's call the first payment "x" dollars. Therefore, the second payment will be "2x" dollars.

To find the present value of the loan, we need to discount the future payments back to the present using the interest rate. The formula for present value is given by:

\[PV = \frac{{CF}}{{(1 + r)^n}}\]

where:
- PV is the present value (amount needed to pay off the loan in the present)
- CF is the future cash flow (the payment amount)
- r is the interest rate as a decimal
- n is the number of time periods

In our case, CF for the first payment is "x" dollars, and CF for the second payment is "2x" dollars. The interest rate, r, is 8.5% or 0.085 as a decimal. The time period, n, will be 3 years for the first payment and 6 years for the second payment.

So, for the first payment, the present value will be:

\[PV_1 = \frac{{x}}{{(1 + 0.085)^3}}\]

And for the second payment, the present value will be:

\[PV_2 = \frac{{2x}}{{(1 + 0.085)^6}}\]

Since Claire plans to pay off the loan in full (borrowed $\$5,\!000$), the sum of the present values will be equal to the loan amount:

\[PV_1 + PV_2 = 5000\]

Substituting the values we have, we can solve for "x":

\[\frac{{x}}{{(1 + 0.085)^3}} + \frac{{2x}}{{(1 + 0.085)^6}} = 5000\]

Now, we can solve this equation to find the value of "x".