Matrices: \(A\neq I\) and \(A^2=I\)

Algebra Level 4

\[A=\frac{1}{5}\left(\begin{array}{cc} -3& a\\b& c \end{array}\right)\qquad \text{and}\qquad \, A^2=\left(\begin{array}{cc} 1& 0\\0& 1 \end{array}\right)\]

Given that \(a, b\) and \(c\) are integers satisfying the constraints above, find the maximum value of \(a+b+c\).

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