\[\large A=\sum_{k=1}^{99} \left( 2\sin ^4 x +2\cos^2 x +\frac { \sin ^2 2x }2\right)^k \]

Find \(\left\lceil \frac { A }{ { 4 }^{ 49 } } \right\rceil \).

\(\)

**Notation:** \( \lceil \cdot \rceil \) denotes the ceiling function.

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