There are \(n\) polynomials \(\left(P_1(x)..... P_n(x)\right)\) satisfying the following two equations:

\[\begin{align} 8 \cdot P(x) &= (P'(x))^2\cdot P''(x) \\ P(0) &= 4. \end{align}\]

Find the value of \(\displaystyle \sum_{i=1}^n P_i(1).\)

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