HMMT November 2022 U34

A random binary string of length 1000 is chosen. Let $L$ be the expected length of its longest (contiguous) palindromic substring. Estimate $L$. An estimate of $E$ will receive $\left\lfloor 20 \min \left(\frac{E}{L}, \frac{L}{E}\right)^{10}\right\rfloor$ points.