\begin{equation*}
\begin{split}
& \text{Let us expand the expression completely using algebraic identities step by step:} \\\\
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& \textbf{Step 1: Rearrange the order of the binomial factors.} \\\\
& \text{The given expression is:} \\\\
& (2xy - z^2)(4x^2y^2 + z^4)(2xy + z^2)(16x^4y^4 + z^8) \\\\
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& \text{Using the commutative property of multiplication, we group the conjugate pairs together:} \\\\
& [(2xy - z^2)(2xy + z^2)] \cdot (4x^2y^2 + z^4) \cdot (16x^4y^4 + z^8) \\\\
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& \textbf{Step 2: Expand the first group using the difference of squares identity.} \\\\
& \text{Apply the identity } (a - b)(a + b) = a^2 - b^2 \text{ where } a = 2xy \text{ and } b = z^2: \\\\
& (2xy)^2 - (z^2)^2 = 4x^2y^2 - z^4 \\\\
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& \textbf{Step 3: Combine this intermediate result with the next binomial factor.} \\\\
& \text{Substitute the result back into the expression:} \\\\
& [(4x^2y^2 - z^4)(4x^2y^2 + z^4)] \cdot (16x^4y^4 + z^8) \\\\
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& \textbf{Step 4: Apply the difference of squares identity a second time.} \\\\
& \text{Using the identity where } a = 4x^2y^2 \text{ and } b = z^4: \\\\
& (4x^2y^2)^2 - (z^4)^2 = 16x^4y^4 - z^8 \\\\
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& \textbf{Step 5: Combine with the final binomial factor and expand one last time.} \\\\
& \text{Substitute the second intermediate result back into the full expression:} \\\\
& (16x^4y^4 - z^8)(16x^4y^4 + z^8) \\\\
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& \text{Apply the difference of squares identity where } a = 16x^4y^4 \text{ and } b = z^8: \\\\
& (16x^4y^4)^2 - (z^8)^2 \\\\
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& \textbf{Step 6: Simplify each squared term.} \\\\
& \bullet \quad (16x^4y^4)^2 = 16^2 \times (x^4)^2 \times (y^4)^2 = 256x^8y^8 \\\\
& \bullet \quad (z^8)^2 = z^{16} \\\\
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& \text{Combining these values yields the final expanded expression:} \\\\
& 256x^8y^8 - z^{16} \\\\
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& \textbf{Conclusion:} \\\\
& \text{Therefore, the completely expanded algebraic form is:} \\\\
& \bbox[5px, border: 2px solid magenta]{256x^8y^8 - z^{16}}
\end{split}
\end{equation*}