Consider a robot world with a robot R in a $4$x$4$ maze. Here are the rules for the robot world:

$$ The maze has walls restricting the movement of the robot.

$$ The robot can face east $($E$),$ west $($W$),$ north $($N$),$ or south $($S$).$

$$ The robot can move to a neighboring cell if there is no wall blocking the robot. It uses the move $($M$)$ operation to change the cell location.

$$ The robot can rotate to change the direction it faces using E$,$ W$,$ N$,$ or S to set its new direction.

$$ The robot has a specific location in the maze.

$$ There are up to three targets $($T$)$ that can be placed in the maze.

$$ The robot must traverse the maze to pick up all targets in the maze.

$$ A maze may not have any targets at all.

$$ Robots may visit a cell with a target to pick up the target.

$$ In the initial state of the system, the robot may not be located in a cell where there is a target and there cannot be more than one target in each cell.

Below is a sample maze with the robot R located in cell $(2,4)$ facing north, denoted as R$($N$),$ and two targets located in cells $(4,1)$ and $(3,3),$ respectively. The north$-$facing robot can change its direction to R$($W$)$ to face west using operation W$.$ The robot can change its location to cell $(3,4).$ However, it must first rotate to face south before executing the move operation.

HW$2$ Q$5$ Maze$-1.$jpg

Answer the following questions about this maze.

Not considering any targets and assuming that the robot world has a single robot located inside the maze, compute the size of the state space, i$.$e$.,$ the number of states of the robot world:

$($enter a number only$)$

A single rotation may rotate the robot from north to south, east to west, or east to north, and so forth.

Including rotations $($e$.$g$.,$ north to south$),$ how many steps does it take for the robot to reach the cell of the closest target:

$($enter a number only$)$