如何通过匹配R中所有可能的组合来从宽到长旋转多个列

我找到了this case和this other，但它们不能完全回答我的问题.

Input data个是宽格式的:

dat0
    # A tibble: 5 × 6
         id gp       a1    a2    a3    a4
      <dbl> <chr> <dbl> <dbl> <dbl> <dbl>
    1     1 A         2    10   150  1000
    2     2 B         4    20   200  3000
    3     3 C         7    30   350  4000
    4     4 D         8    40   400  7000
    5     5 E         9    50   700  8000  

dat0 <-
structure(list(id = c(1, 2, 3, 4, 5), gp = c("A", "B", "C", "D", 
"E"), a1 = c(2, 4, 7, 8, 9), a2 = c(10, 20, 30, 40, 50), a3 = c(150, 
200, 350, 400, 700), a4 = c(1000, 3000, 4000, 7000, 8000)), class = c("tbl_df", 
"tbl", "data.frame"), row.names = c(NA, -5L)) 

如何将多个列透视为两个列ax和ay，以便:

• 表示初始列a1至a4的所有组合，
• 两个初始的第一列(这里是id和gp，作为例子给出)被复制与可能的组合一样多的次数，
• 创建新的列ax_vs_ay以对组合进行分类.

Desired output data:

print(dat1, n=30)
# A tibble: 30 × 5
      id gp    ax_vs_ay    ax    ay
   <dbl> <chr> <chr>    <dbl> <dbl>
 1     1 A     a1_vs_a2     2    10
 2     2 B     a1_vs_a2     4    20
 3     3 C     a1_vs_a2     7    30
 4     4 D     a1_vs_a2     8    40
 5     5 E     a1_vs_a2     9    50
 6     1 A     a1_vs_a3     2   150
 7     2 B     a1_vs_a3     4   200
 8     3 C     a1_vs_a3     7   350
 9     4 D     a1_vs_a3     8   400
10     5 E     a1_vs_a3     9   700
11     1 A     a1_vs_a4     2  1000
12     2 B     a1_vs_a4     4  3000
13     3 C     a1_vs_a4     7  4000
14     4 D     a1_vs_a4     8  7000
15     5 E     a1_vs_a4     9  8000
16     1 A     a2_vs_a3    10   150
17     2 B     a2_vs_a3    20   200
18     3 C     a2_vs_a3    30   350
19     4 D     a2_vs_a3    40   400
20     5 E     a2_vs_a3    50   700
21     1 A     a2_vs_a4    10  1000
22     2 B     a2_vs_a4    20  3000
23     3 C     a2_vs_a4    30  4000
24     4 D     a2_vs_a4    40  7000
25     5 E     a2_vs_a4    50  8000
26     1 A     a3_vs_a4   150  1000
27     2 B     a3_vs_a4   200  3000
28     3 C     a3_vs_a4   350  4000
29     4 D     a3_vs_a4   400  7000
30     5 E     a3_vs_a4   700  8000  

谢谢你的帮助