Let $a,b,c,d$ be non-negative integers. $(1)$ If $a^2+b^2-cd^2=2022 ,$ find the minimum of $a+b+c+d;$ $(1)$ If $a^2-b^2+cd^2=2022 ,$ find the minimum of $a+b+c+d .$
Source: Jianxi jian
Tags: inequalities, algebra, number theory
Let $a,b,c,d$ be non-negative integers. $(1)$ If $a^2+b^2-cd^2=2022 ,$ find the minimum of $a+b+c+d;$ $(1)$ If $a^2-b^2+cd^2=2022 ,$ find the minimum of $a+b+c+d .$
