How to prove inequaltiy for real numbers [closed]

Let $a_1, a_2, a_3, a_4, a_5 \in \mathbb{R}$. Suppose $a_1, a_2, a_3, a_4, a_5 > 0$. Suppose $a_1 + a_2 + a_3 + a_4 + a_5 = 1$

Prove $$\big( \frac{1}{a_1} - 1 \big)\big( \frac{1}{a_2} - 1 \big)\big( \frac{1}{a_3} - 1 \big)\big( \frac{1}{a_4} - 1 \big)\big( \frac{1}{a_5} - 1 \big) \geq 1024$$

I'm trying to figure out if there's a good way to go about doing this proof without expanding everything out. Perhaps using GM or AM, but I'm not entirely sure where it comes to play. I think I'd also use the fact that $4^5 = 1024$, but again not entirely sure how.

Any advice on this question would be much appreciated!

Hint:

Multiply both sides by $a_1a_2a_3a_4a_5$: $$(1-a_1)(1-a_2)(1-a_3)(1-a_4)(1-a_5)\ge 4^5a_1a_2a_3a_4a_5$$

Now use AM-GM: $1-a_1=a_2+a_3+a_4+a_5\ge 4\sqrt[4]{a_2a_3a_4a_5}$, etc...