Inspired by Yash Singhal

Let $a$ and $b$ be the roots of the quadratic equation $x^2-(k-2)cx-(k-1)d=0$

Let $c$ and $d$ be the roots of the quadratic equation $x^2-(k-2)ax-(k-1)b=0$

If $a,b,c$ and $d$ are distinct, non zero real numbers such that

$\large{a+b+c+d=100}$

Find the integral value of $\large k$ .

Inspirations - 1 and 2