Cash flow $1$ consists of three payments: $A$ in $5$ periods, $B$ in $10$ periods and $C$ in $15$ periods. Cash flow $2$ consists of $2$ payments: $x$ in $6$ periods and $Y$ in $12$ periods. As of now, at yield $j$ per period for all maturities, the two sets of cashflows have the same present value and the same Macaulay duration. Suppose that one period from now the yield is still $j$ for all maturities. Show that

$($a$)$ the two sets of cash flows will have the same present value;

$($b$)$ the two sets of cash flows have the same Macaulay duration at that time; and

$($c$)$ the Macaulay duration one period from now is exactly $1$ less than it is now.