Bond valuation:

$(1)$ Suppose there is a $10-$year, $$1000$ bond with an $8\%$ coupon rate and semiannual coupons. If the bond's yield to maturity is $7\%$ APR, what is the bond's price?

$(2)$ Assuming the yield to maturity remains constant, what is the price of the bond immediately before it makes its first coupon payment?

$(3)$ Assuming the yield to maturity remains constant, what is the price of the bond immediately after it makes its first coupon payment?