Colors of different nature can be produced by mixing three fundamental colors that are red, green and blue. Consider you are assigned a task to design a digital system that is equipped with three different LEDs and four different color sensors. One LED is red (symbolized with R), one is green (symbolized with G) and one is blue (symbolized with L) while one sensor detects white color (symbolized with W), one detects black color (symbolized with B), one detects cyan color (symbolized with C) and one sensor detects magenta color (symbolized with M).

It is known that each color sensor produces logic 1 output if and only if when an object of its corresponding color passes in front of it, otherwise logic 0. Each LED turns on when logic 0 is provided to it and remains turned off otherwise. Three LEDs are to be turned on according to the following conditions.

i. If only white color is detected then all three LEDs should be on

ii. If only black color is detected then all three LEDs should be off

iii. If both cyan and white are detected then only blue and green LEDs should be on

iv. If both magenta and white are detected then only blue and red LEDs should be on

v. If only magenta is detected then only red LED should be on

vi. If only cyan is detected then only blue LED should be on

vii. LEDs remain off for any other outputs from all sensors

In order to implement this digital system, you are required to provide:

i. Truth table for controlling three LEDs

ii. K-map derivation of Boolean functions

iii. A single NAND-only circuit controlling three LEDs