Distinct Symbols in a Palindromic Access Code

A 5-character access code is formed by choosing each character independently and uniformly from $\{A, B, C, D, E, F\}$ (repetition allowed). The code is then accepted only if it is a palindrome (reads the same forwards and backwards). Among all palindromic codes, find $P(\text{at least 3 distinct symbols appear in the code})$.