Given an initial sequence $a_1, a_2, \ldots, a_n$ of real numbers, we perform a series of steps. 
At each step, we replace the current sequence $x_1,x_2,\ldots,x_n$
with the sequence $|x_1-a|,|x_2-a|,\ldots,|x_n-a|$ for some number $a$ that can vary with each step.

(a) Prove that no matter what sequence we start with, there is some way to do steps to reach the sequence $0,0,\ldots,0$.

(b) * Determine (with proof) the maximum, over all sequences of length $n$, of the minimum number of steps required to reach $0,0,\ldots,0$.