SHARYGIN 2022 Problem 14

A triangle $ABC$ is given. Let $C'$ and $C'_{a}$ be the touching points of sideline $AB$ with the incircle and with the excircle touching the side $BC$ . Points $C'_{b}$ , $C'_{c}$ , $A'$ , $A'_{a}$ , $A'_{b}$ , $A'_{c}$ , $B'$ , $B'_{a}$ , $B'_{b}$ , $B'_{c}$ are defined similarly. Consider the lengths of $12$ altitudes of triangles $A'B'C'$ , $A'_{a}B'_{a}C'_{a}$ , $A'_{b}B'_{b}C'_{b}$ , $A'_{c}B'_{c}C'_{c}$ . (a) (8-9) Find the maximal number of different lengths. (b) (10-11) Find all possible numbers of different lengths.