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Max and Min

Source: 1986 National High School Mathematics League, Exam One, Problem 10

February 24, 2020

Problem Statement

x,y,zx,y,z are nonnegative real numbers, and 45x+9y+4z−68×25x+9y+4z+256=04^{\sqrt{5x+9y+4z}}-68\times2^{\sqrt{5x+9y+4z}}+256=0. Then, the product of the maximum and minimum value of x+y+zx+y+z is________.
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