The General Form of the Non$-$linear Optimization Problem:

$min{x}_1{x}_4(x_1+x_2+x_3)+x_3$

$s.t.x_1{x}_2{x}_3{x}_{4}25$

$x_1^2+x_2^2+x_3^2+x_4^2=40$

$1x_1,x_2,x_3,x_{4}5$

$x_0=(1,5,5,1)$

This problem has a nonlinear objective that the optimizer attempts to minimize. The variable values at the optimal solution are subject to both equality and inequality constraints. The product of the four variables must be greater than $25$ while the sum of squares of the variables must also equal $40.$ In addition, all variables must be between $1$ and $5$ and the initial guess is $1=1,x2=5,x3=5,$ and $x4=1.$ Write a Python program that solves the problem above.