What is the value of \(x\)

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\[\begin{aligned} &x+\frac{x}{2}+\frac{x}{4}+\cdots+\frac{x}{512}+\frac{x}{1024}=1+2+4+\cdots+512+1024 \\\\ &\Leftrightarrow \quad x\left(1+\frac{1}{2}+\frac{1}{2^{2}}+\cdots+\frac{1}{2^{9}}+\frac{1}{2^{10}}\right)=1+2+2^{2}+\cdots+2^{9}+2^{10} \\\\ &\Leftrightarrow \quad x \cdot\left[1 \cdot \frac{1-\left(\frac{1}{2}\right)^{11}}{1-\frac{1}{2}}\right]=1 \cdot \frac{1-2^{11}}{1-2} \\\\ &\Leftrightarrow \quad 2 x \cdot\left(1-\frac{1}{2^{11}}\right)=2^{11}-1 \\\\ &\Leftrightarrow \quad 2 x \cdot \frac{2^{11}-1}{2^{11}}=2^{11}-1 \\\\& \Leftrightarrow x=\frac{\left(2^{11}-1\right) \cdot 2^{11}}{2 \cdot\left(2^{11}-1\right)}=2^{10}=1024 \end{aligned}\]
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