Two-Step Tower in an Additive Bernoulli Markov Chain

Let $X_1 \sim \operatorname{Uniform}\{0, 1\}$. Given $X_1$, let $X_2 = X_1 + B_1$ where $B_1 \sim \operatorname{Bernoulli}(1/2)$ independent of $X_1$. Given $X_2$, let $X_3 = X_2 + B_2$ where $B_2 \sim \operatorname{Bernoulli}(1/2)$ independent of everything else.

(a) Using the tower property $E[X_3 \mid X_1] = E[E[X_3 \mid X_2] \mid X_1]$, find $E[X_3 \mid X_1]$ and $E[X_3]$.

(b) Using Eve's law, find $\operatorname{Var}(X_3)$.