Bird in the Hand vs Discounted Wait

You face two periods. In period 1 a reward X1 ~ Uniform(0,1) is offered; accept it now to receive X1, or wait. If you wait, in period 2 you must accept X2 ~ Uniform(0,1), but a reward received in period 2 is worth only a fraction beta = 0.8 of its face value (discounting). No recall. Find the optimal period-1 acceptance threshold and the expected payoff of the optimal policy.