Thread: please help with this Q...... of Finite automaton

let L be a regular language of binary strings.

let L* be another language such that (x:x belong to L and x ^ R belong to L) where x ^ R is the reverse

show that L* is also regular

The idea here is to make a new automaton which computes and simultaneously, such that it will enter an accepting state iff both and enter an accept state at the same point of the computation. You complete the details.