In a mile race between A and B, the time it takes A to complete the mile is an exponential random variable with rate λa and is independent of the time it takes B to complete the mile, which is an exponential random variable with rate λb. The one who finishes earliest is declared the winner and receives Re

−αt if the winning time is t, where R and α are constants. If the loser receives 0, find the expected amount that runner A wins.