Two Permutations (Hard Version)

给定两个排列 $p, q$，长度分别为 $n, m$。对一个长为 $t$ 的排列你可以按如下方式进行一次参数为 $i$ 的操作（$1\le i \le t$）：把序列划分成 $i$ 以前、$i$ 这个位置和 $i$ 以后三个段（段可能为空），接着交换一、三两个段。在每一轮中，你必须选择一对整数 $i, j$ 使得 $1\le i\le n, 1\le j\le m$，然后对 $p$ 用参数 $i$ 进行操作，对 $q$ 用 $j$ 进行操作。你的目标是使得两个排列都有序。输出最小轮数和一种操作方案。无解输出一行一个 $-1$。

This is the hard version of the problem. The difference between the two versions is that you have to minimize the number of operations in this version. You can make hacks only if both versions of the problem are solved. You have two permutations $ ^{\dagger} $ $ p_{1}, p_{2}, \ldots, p_{n} $ (of integers $ 1 $ to $ n $ ) and $ q_{1}, q_{2}, \ldots, q_{m} $ (of integers $ 1 $ to $ m $ ). Initially $ p_{i}=a_{i} $ for $ i=1, 2, \ldots, n $ , and $ q_{j} = b_{j} $ for $ j = 1, 2, \ldots, m $ . You can apply the following operation on the permutations several (possibly, zero) times. In one operation, $ p $ and $ q $ will change according to the following three steps: - You choose integers $ i $ , $ j $ which satisfy $ 1 \le i \le n $ and $ 1 \le j \le m $ . - Permutation $ p $ is partitioned into three parts using $ p_i $ as a pivot: the left part is formed by elements $ p_1, p_2, \ldots, p_{i-1} $ (this part may be empty), the middle part is the single element $ p_i $ , and the right part is $ p_{i+1}, p_{i+2}, \ldots, p_n $ (this part may be empty). To proceed, swap the left and the right parts of this partition. Formally, after this step, $ p $ will become $ p_{i+1}, p_{i+2}, \ldots, p_{n}, p_{i}, p_{1}, p_{2}, \ldots, p_{i-1} $ . The elements of the newly formed $ p $ will be reindexed starting from $ 1 $ . - Perform the same transformation on $ q $ with index $ j $ . Formally, after this step, $ q $ will become $ q_{j+1}, q_{j+2}, \ldots, q_{m}, q_{j}, q_{1}, q_{2}, \ldots, q_{j-1} $ . The elements of the newly formed $ q $ will be reindexed starting from $ 1 $ . Your goal is to simultaneously make $ p_{i}=i $ for $ i=1, 2, \ldots, n $ , and $ q_{j} = j $ for $ j = 1, 2, \ldots, m $ . Find any way to achieve the goal using the minimum number of operations possible, or say that none exists. Please note that you have to minimize the number of operations. $ ^{\dagger} $ A permutation of length $ k $ is an array consisting of $ k $ distinct integers from $ 1 $ to $ k $ in arbitrary order. For example, $ [2,3,1,5,4] $ is a permutation, but $ [1,2,2] $ is not a permutation ( $ 2 $ appears twice in the array), and $ [1,3,4] $ is also not a permutation ( $ k=3 $ but there is $ 4 $ in the array).

The first line contains two integers $ n $ and $ m $ ( $ 1 \le n, m \le 2500 $ ). The second line contains $ n $ integers $ a_1, a_2, \ldots, a_n $ ( $ 1 \le a_i \le n $ ). The third line contains $ m $ integers $ b_1, b_2, \ldots, b_m $ ( $ 1 \le b_i \le m $ ). It is guaranteed that $ a $ and $ b $ are permutations.

If there is no solution, print a single integer $ -1 $ . Otherwise, print an integer $ k $ — the number of operations to perform, followed by $ k $ lines, each containing two integers $ i $ and $ j $ ( $ 1 \le i \le n $ , $ 1 \le j \le m $ ) — the integers chosen for the operation. If there are multiple solutions, print any of them. Please note that you have to minimize the number of operations.

3 5
2 1 3
5 2 1 4 3

2
3 4
2 4

4 4
3 4 2 1
2 4 1 3

3
3 3
1 4
4 2

2 2
1 2
2 1

-1

In the first test case, we can achieve the goal within $ 2 $ operations: 1. In the first operation, choose $ i = 3 $ , $ j = 4 $ . After this, $ p $ becomes $ [3, 2, 1] $ and $ q $ becomes $ [3, 4, 5, 2, 1] $ . 2. In the second operation, choose $ i = 2 $ , $ j = 4 $ . After this, $ p $ becomes $ [1, 2, 3] $ and $ q $ becomes $ [1, 2, 3, 4, 5] $ . In the third test case, it is impossible to achieve the goal.