A point has rectangular coordinates $(2,-1,-2)$ and spherical coordinates $(\rho, \theta, \phi).$  Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, 2 \phi).$
<|im_start|>system
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<|im_start|>user
A point has rectangular coordinates $(2,-1,-2)$ and spherical coordinates $(\rho, \theta, \phi).$  Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, 2 \phi).$
Please reason step by step, and put your final answer within \boxed{{}}.<|im_end|>
<|im_start|>assistant

user
\left( -\frac{8}{3}, \frac{4}{3}, -\frac{1}{3} \right)
A point has rectangular coordinates $(2,-1,-2)$ and spherical coordinates $(\rho, \theta, \phi).$  Find the rectangular coordinates of the point with spherical coordinates $(\rho, \theta, 2 \phi).$
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