plotting - Calculate and plot the tangent hyperplane to a function of three variables

f:= x^3 + y*x*z - z^2 + x*z

and if {{1,2,1},{2,7,8},{6,0,2},{9,1,1}} are points, how can I find the tangent plane to f at the four points in the same time.

I saw some codes to calculate the tangent , for example,

tg[f_, x_, p_] := (f'[x] /. x -> p) (x - p) + f[p]
q = #^3 - 3 #^2 &;
Manipulate[
 Plot[{q[x], tg[q, u, m] /. u -> x}, {x, -3, 3}, PlotRange -> {-5, 5},
   Epilog -> {Red, PointSize[0.02], Point[{m, q[m]}]}], {m, -1, 1, 
  0.01}]

but I am confused this did not answer my question. Thanks for any help.

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