3D points challenge

Geometry Level 3

The points \(A\), \(B\), \(P\), and \(Q\) all lie on one line, with \(A=(-3, 5, 8).\)

Point \(P\) lies in the \(xy\)-plane between \(A\) and \(B\) such that the distance from \(A\) to \(P\) is twice the distance from \(P\) to \(B\). Furthermore, \(Q\) sits on the \(z\)-axis such that the distance from \(A\) to \(Q\) is twice the distance from \(Q\) to \(B\).

If \(B=(x,y,z),\) what is the value of \(x+y+z?\)

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