Combine a disc and a weighted die

An unbiased disc has a single dot marked on one side and two dots marked on the other side. A tetrahedral die has faces marked with score of 1, 2, 3, and 4. The probability of getting a score of 1, 2, 3, and 4 is $\frac15,p,\frac15$ and $q$ respectively, where $p,q\in[0,1]$.

A game is played by throwing the disc and the die together. The random variable $S$ is the sum of the score showing on the die and twice the number of dots showing on the disc.

(i) Find $P(S=6)$.

Given that $P(S=4)=\frac16$,

(ii) calculate the values of p and q, and

(iii) find the probability distribution of S.