KMO 2018 Problem 2

For a positive integer $n$ , denote $p(n)$ to be the number of nonnegative integer tuples $(x,y,z,w)$ such that $x+y+2z+3w=n-1$ . Also, denote $q(n)$ to be the number of nonnegative integer tuples $(a,b,c,d)$ such that

(i). $a+b+c+d=n$ . (ii). $a \ge b$ , $c \ge d$ , $a \ge d$ . (iii). $b < c$ .

Prove that for all $n$ , $p(n) = q(n)$ .