Number Theory or Algebra? Part 10 - Finale

Find the number of tri-ordinates (x,y,z)(x, y, z), in which x,yx, y and zz are all positive integers, which satisfy the equation

2x+334y+5+5=z22^{x +3} - 3\cdot4^{y+5} + 5 = z^{2}

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