Let \( f:\mathbb{N}\to \mathbb{R} \) be a function such that \( f(x+y)=2f(x)f(y) \) for natural numbers \( x \) and \( y \). If \( f(1)=2 \), then the value of \( \alpha \) for which \( \displaystyle \sum_{k=1}^{10} f(\alpha+k)=\dfrac{512}{3}\left(2^{20}-1\right) \) holds, is:
