Question $3$: Complete a Path

Shaanan wants to make sure the grass on his lawn is of equal length. It means that each time Shaanan mows, he goes through a path covering the entire backyard with his lawnmower. Also, given he has to mow the entire lawn, he really wants to avoid any repetition $($i$.$e$.,$ mowing the same piece of grass twice$).$

He comes up with two strategies to mow the lawn that will satisfy his requirement above:

Strategy $1-$ Back and Forth Path: start at the top left corner and go straight to the right. Every time he reaches the edge of the backyard, make a U$-$turn and continue mowing.

Strategy $2-$ Spiral Path: start at the top left corner and go clockwise through a spiral path until he reaches the centre with no more land to cover.

Your Task

Write a function complete$\_$path$($dimension$,$ path$\_$type$).$

Input:

dimension: A $2-$element tuple $($row$\_$dim, col$\_$dim$)$ signalling the dimension of the backyard $($described in Q$2).$

path$\_$type: A str with value of either "back$\_$and$\_$forth" or "spiral", signalling the type of the path.

Output: The path as described above, depending on path$\_$type. The format of the path is the same as in Q$2.$ The returned path should contain every coordinate in the backyard, without duplication.

Assumptions you can make:

The inputs are well$-$formatted with the correct type.

The dimension tuple contains two int with positive values.

Example calls:

$>>>$ print$($complete$\_$path$((3,4),$ "back$\_$and$\_$forth"$))$

$[(0,0),(0,1),(0,2),(0,3),(1,3),(1,2),(1,1),(1,0),(2,0),(2,1),(2,2),(2,3)]$

$>>>$ print$($complete$\_$path$((3,4),$ "spiral"$))$

$[(0,0),(0,1),(0,2),(0,3),(1,3),(2,3),(2,2),(2,1),(2,0),(1,0),(1,1),(1,2)]$