Gita starts from point A and walks 1000 m east. She then walks 800 m north, followed by 640 m west and 512 m south, reaching point B. After this, she continues moving in the same cyclic order: east, north, west, south, with each successive movement 20% shorter than the previous one. After infinitely many such moves, approximately how far in meters will Gita be from her starting point A?

Each move is $0.8$ times the previous one, cycling through East, North, West, South. The first four moves are:

East $= 1000$, North $= 800$, West $= 640$, South $= 512$

After every four moves, the pattern repeats, scaled by $(0.8)^4 = 0.4096$.

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So each direction's distances form a geometric progression with common ratio $0.4096$, and the common denominator for each sum is:

$1 - 0.4096 = 0.5904$

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Net eastward displacement (East minus West):

$\dfrac{1000 - 640}{0.5904} = \dfrac{360}{0.5904}$

Net northward displacement (North minus South):

$\dfrac{800 - 512}{0.5904} = \dfrac{288}{0.5904}$

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Distance from $A$:

$d = \dfrac{\sqrt{360^2 + 288^2}}{0.5904}$

$360^2 + 288^2 = 129600 + 82944 = 212544$

$\sqrt{212544} \approx 461.02$

$d \approx \dfrac{461.02}{0.5904} \approx 780.87$

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Gita is approximately $781$ m from $A$.

Answer $= 781$