Let $a_1,\ldots,a_n \in [0,1)$ and $P:=(1-a_1)(1-a_2)\ldots(1-a_n)$. Find good lower-bound for $P$ in terms of $\|a\|_\infty$ and $\|a\|_2$

Question

Let $a_1,\ldots,a_n \in [0,1)$, where $n>0$ is a large integer. Define $\|a\|_\infty := \max(a_1,a_2,\ldots,a_n)$, $\|a\|_2 := \sqrt{a_1^2 + a_2^2+\ldots + a_n^2}$, and $P:=(1-a_1)(1-a_2)\ldots(1-a_n)$.

Question. Is there a better (larger is better) lower-bound for $P$ as a function of $\|a\|_\infty$ and $\|a\|_2$, apart from the trivial one $P \ge (1-\|a\|_\infty)^n$ ?