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Append a z-axis to the 2-dimensional plane and conquer the realm of 3-dimensional space.
If the two spheres \((x-4)^2+(y-1)^2+z^2=27\) and \((x-13)^2+(y-10)^2+(z+9)^2=a\) are externally tangent to each other, what is the value of \(a?\)
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Consider a sphere that is tangent to the \(xy\)-, \(yz\)-, and \(xz\)-planes. If the sphere passes through \((6,3,-3),\) what is the product of all distinct possible values of the sphere's radius.
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The center of the sphere \[x^2-10x+y^2+z^2-18z+99=0\] is \((a,b,c).\) Find the value of \(a+b+c.\)
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The equation of a sphere centered at \((4,-6,-5)\) with radius \(2\) is \[x^2+y^2+z^2+ax+by+cz+d=0.\] Find the value of \(a+b+c+d.\)
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What is the radius of the sphere \[x^2+y^2+z^2=22?\]
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