Replace the letters $a$, $b$, $c$ and $d$ in the following expression with the numbers $1$, $2$, $3$ and $4$, without repetition:
$a+\cfrac{1}{b+\cfrac{1}{c+\cfrac{1}{d}}}$
Find the difference between the maximum value and the minimum value of the expression.The answer is in the form \frac{m}{n}, where gcd(m, n) = 1. Please provide the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
user