$(15$ points$)$ One hundred heart$-$shaped balloons are placed in a park. Forty of the balloons are of red color, thirty of them are of green color, twenty of them are of blue color, and ten of them are of white color.

$($a$)$ How many balloons do we need to collect to guarantee that four balloons are of the same color? $($Steps required.$)$

$($b$)$ How many balloons do we need to collect to guarantee that we have four balloons of each color? $($Steps required.$)$