HMMT November 2022 U35

For each $i \in\{1, \ldots, 10\}, a_{i}$ is chosen independently and uniformly at random from $\left[0, i^{2}\right]$. Let $P$ be the probability that $a_{1}<a_{2}<\cdots<a_{10}$. Estimate $P$. An estimate of $E$ will earn $\left\lfloor 20 \min \left(\frac{E}{P}, \frac{P}{E}\right)\right\rfloor$ points.