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To find the Taylor series expansion of the function $f(x, y, z) = x y z$ around the point $(1, 1, 1)$ using Taylor's approach, we first recall the formula for the multivariable Taylor series: $$ f(x, y, z) \approx f(a, b, c) + f_x(a, b, c)(x-a) + f_y(a, b, c)(y-b) + f_z(a, b, c)(z-c) + \frac{1}{2!} \Big[ f_{xx}(a, b, c)(x-a)^2 + f_{yy}(a, b, c)(y-b)^2 + f_{zz}(a, b, c)(z-c)^2 + 2f_{xy}(a, b, c)(x-a)(y-b) __________ _____ ________ _________ ______.
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