Shaanan starts from the top left corner as before $($will move to the right$),$ Yige starts from the top right corner $($will move downwards$),$ Han starts from the bottom right corner $($will move to the left$),$ and Rose starts from the bottom left corner $($will move upwards$).$ No piece of lawn can be repeatedly mowed.

They start mowing at the time step $0.$ Then at each time step, each of the four people completes one of the three actions with their lawnmower:

$*$ Move one step forward without turning $($i$.$e$.,$ no change in direction$).$

$*$ Turn clockwise and move one step forward. This action will only be selected if:

$*$ moving one step forward without turning will result in the person immediately land on a coordinate that has already been mowed, AND

$*$ after turning clockwise and move one step forward, the person immediately lands on a coordinate that has not been mowed$)$

$*$ Do nothing. Select this action if the other two actions are not possible.

During each time step, Shaanan performs his action first, then Yige, then Han, then Rose.

Your Task

Write a function quad$\_$clockwise$\_$spiral$\_$paths$($dimension$,$ time$\_$step$).$

Input:

$*$ dimension: row$\_$dim, col$\_$dim

$*$ time$\_$step: a non$-$negative int signalling the time step when mowing

Output: The function returns a list containing $4$ paths, in the order of Shaanan$$s$,$ Yige$$s$,$ Han$$s and Rose$$s$,$ at the time$\_$step given.

Assumptions you can make:

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

$*$ For this question, you can assume that in the dimension tuple $($row$\_$dim, col$\_$dim$),$ row$\_$dim and col$\_$dim are no less than $2.$