The function $f(x, y)=4 x+4 y$ has an absolute maximum value and absolute minimum value subject to the constraint $25 x^{2}-25 x y+25 y^{2}=25$. Use Lagrange multipliers to find these values. The absolute maximum value is (Type an exact answer.)

Step 1 ：The given function is \(f(x, y)=4 x+4 y\). The constraint is \(g(x, y)=25 x^{2}-25 x y+25 y^{2}-25=0\).

Step 2 ：We use the method of Lagrange multipliers. The Lagrange function is \(L(x, y, \lambda)=f(x, y)-\lambda g(x, y)\).

Step 3 ：So, \(L(x, y, \lambda)=4 x+4 y-\lambda (25 x^{2}-25 x y+25 y^{2}-25)\).

Step 4 ：The partial derivatives of \(L\) are: \(\frac{\partial L}{\partial x}=4-\lambda (50 x-25 y)\), \(\frac{\partial L}{\partial y}=4-\lambda (50 y-25 x)\), and \(\frac{\partial L}{\partial \lambda}=-(25 x^{2}-25 x y+25 y^{2}-25)\).

Step 5 ：Setting these partial derivatives equal to zero gives us the system of equations: \(4-\lambda (50 x-25 y)=0\), \(4-\lambda (50 y-25 x)=0\), and \(25 x^{2}-25 x y+25 y^{2}-25=0\).

Step 6 ：Solving this system of equations, we get two solutions: \((x, y, \lambda)=(1, 1, 0.16)\) and \((x, y, \lambda)=(-1, -1, 0.16)\).

Step 7 ：Substituting these solutions into the function \(f(x, y)\), we get \(f(1, 1)=8\) and \(f(-1, -1)=-8\).

Step 8 ：Therefore, the absolute maximum value is \(\boxed{8}\) and the absolute minimum value is \(\boxed{-8}\).