A teacher decides to play a game with her three students. She writes four positive (real) numbers on the board; $a,b,c,d$.

One student's task is to find the sum of the cube of each number. (i.e. $a^3+b^3+c^3+d^3$)

Another student's task is to find the sum of the reciprocal of each number.

The final student's task is to find the sum of the squared reciprocal of each number.

When they are all finished doing this fun calculation, they multiply their answers together. What is the minimum value of their final answer?