HMMT 2025 C8

Albert writes 2025 numbers $a_{1}$ , ..., $a_{2025}$ in a circle on a blackboard. Initially, each of the numbers is uniformly and independently sampled at random from the interval $[0,1]$ . Then, each second, he simultaneously replaces $a_{i}$ with $\max (a_{i - 1}, a_{i}, a_{i + 1})$ for all $i = 1, 2, \ldots , 2025$ (where $a_{0} = a_{2025}$ and $a_{2026} = a_{1}$ ). Compute the expected value of the number of distinct values remaining after 100 seconds.