\[ \begin{equation} \begin{split} \log_{|m|+2} (a^2&+1) + \log_{|n|+3} (b^2-2b+2) +\\ & +\log_{|p|+4} (a^2+2ac+c^2+1) = 0 \end{split} \end{equation} \]

Enter the number of elements of the set \(A=\{a,b,c,a+b\}\).

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