A Skip-Counting Sequence

A sequence $\{a_n\}$ has $a_1 = 1$, and for $n \geqslant 2$,

Let $b_m$ be the number of terms of $\{a_n\}$ lying in $\left[2^m, 2^{m+1}\right)$, for $m \in \mathbb{N}^*$. Determine, with proof, which of the following are true:

1. $b_3 = 10$; 2. $\dfrac{3n}{4} \leqslant a_n \leqslant \dfrac{3n+3}{4}$; 3. $\displaystyle\sum_{k=1}^{4n} a_k = 6n^2 + 3n$; 4. $\displaystyle\sum_{k=1}^{2n} b_k = \dfrac{4}{3}\left(4^n - 1\right)$.