Electrification

Ellen is a technologist at Advanced Circuit Mastership, currently designing an electrical scheme for a light management system in a luxury hotel room.

The scheme will be embedded in the ceiling of the room. It consists of a rectangular grid, where each cell contains one element. Below is a list of all possible elements, their one-character representations, and descriptions:

'.' — empty cell; no input or output signals;

'/' — wire; takes input signal from the lower left corner and transmits it to the upper right corner;

'\' — wire; takes input signal from the upper left corner and transmits it to the lower right corner;

'X' — intersection; takes input signals from the upper left and lower left corners and transmits them to the lower right and upper right corners, respectively;

'L' — wire splitter; takes input signal from the lower left corner and transmits it to both the upper left and lower right corners;

'7' — exclusive or; takes input signals from the upper left and lower left corners and transmits their XOR to the upper right corner.

The hotel room has $n$ switches and $m$ lamps. Switches are connected to the $n$ topmost positions on the left edge of the scheme, while the lamps are connected to the $m$ topmost positions on the right edge. Ellen's task is to create a scheme connecting the switches and lamps, following these design guidelines:

Room designers assign each lamp a set of switches. A lamp must change its state (toggle on/off) each time a switch assigned to it is flipped. A lamp must not change state when a non-assigned switch is flipped.

The following wiring rules must be satisfied:

• The lower left corner of an 'L' element must have exactly one input wire. The other two corners each must have exactly one output wire.

• The upper right corner of a '7' element must have exactly one output wire. Each of the other two corners must have exactly one input wire.

• Each switch must be connected to exactly one output wire.

• Each lamp must be connected to exactly one input wire.

• Any left-side position not connected to a switch, and any right-side position not connected to a lamp, must have no input or output wires.

• Each other corner must have either no input and no output wires, or exactly one input and one output wire.

Note that the set of elements is designed so that cyclic dependencies cannot occur.

Ellen receives the switch-to-lamp assignment. Help her design the required scheme.