Divisible Sequences

Let $m$ be a positive integer, and let $a_1, a_2, \dots, a_{4m+2}$ be an arithmetic sequence with nonzero common difference. If after deleting two terms $a_i$ and $a_j$ ($i < j$) the remaining $4m$ terms can be evenly divided into $m$ groups of $4$, each of which forms an arithmetic sequence, then the sequence is called $(i, j)$-divisible.

1. Write down all pairs $(i, j)$ with $1 \leqslant i < j \leqslant 6$ for which $a_1, a_2, \dots, a_6$ is $(i, j)$-divisible. 2. For $m \geqslant 3$, prove that $a_1, a_2, \dots, a_{4m+2}$ is $(2, 13)$-divisible. 3. Two numbers $i < j$ are chosen at random from $1, 2, \dots, 4m+2$; let $P_m$ be the probability that the sequence is $(i, j)$-divisible. Prove that $P_m > \dfrac{1}{8}$.