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Algebra Level 4

\[ \begin{array}{rl} P + A &= AP\\ P + N &= NP\\ AP + NP &= PNAP \end{array} \] Determine all complex solutions of the system of equations above. If your set of solutions are of the following form \(\left(P_1, A_1, N_1\right), \left(P_2, A_2, N_2\right), \dots\), input the real value of the following sum as your answer \[\sum\limits_{i=1}^{\infty} P_iP_iA_iP_i\]

Clarification: The concatenation of terms indicates multiplication of variables. For instance, \(AP\) resembles the product of \(A\) and \(P\).


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