Let $\oplus$ denote the xor binary operation. Define $x \star y = (x+y) - (x \oplus y).$ Compute 
\[ \sum_{k=1}^{63} (k \star 45). \]

**Remark:** The xor operation works as follows: when considered in binary, the $k$th binary digit of $a \oplus b$ is $1$ exactly when the $k$th binary digits of $a$ and $b$ are different. For example, $5 \oplus 12 = 0101_2 \oplus 1100_2 = 1001_2 = 9.$

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