Inspired by Fixed-Point Theorem Story

Suppose there are two functions in $xyz$ defined on a unit-sphere centered on the origin.

$f = 3 x + 5 y^3 + 7 z^5 + 1 \\ g = x y^2 + z - 3$

There are two diametrically opposed points on the sphere, such that the function $f$ has the same value at both points, and the function $g$ has the same value at both points. If one of these points has coordinates $(x',y',z')$ , enter your answer as $| x' + y' + z' |$ , where $| \cdot |$ denotes absolute value.

Inspired by this account from Marvin Minsky regarding his undergrad thesis work. Further problems may explore more elaborate extensions of this idea.