Let $\left\{b_{n}\right\}_{n=0}^{\infty}$ be a sequence of positive real numbers such that $b_{0}=1$, $b_{n}=2+\sqrt{b_{n-1}}-2 \sqrt{1+\sqrt{b_{n-1}}}$. Calculate the sum of $b_{n} 2^{n}$ from $n=1$ to $\infty$. The original answer includes a constant term and a logarithmic term. Please provide the integer part of the answer.

Let's think step by step and output the final answer within \boxed{}.
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