On the intimate association between even binary palindromic words and the Collatz-Hailstone iterations

The celebrated $3x+1$ problem is reformulated via the use of an analytic expression of the trailing zeros sequence resulting in a single branch formula $f(x)+1$ with a unique fixed point. The resultant formula $f(x)$ is also found to coincide with that of the discrete derivative of the sorted sequence of fixed points of the reflection operator on even binary palindromes of fixed even length 2k in any interval [0⋯2²ᵏ-1]. A set of equivalent reformulations of the problem are also presented.