Euler Problem 220 Heighway Dragon or Dragon Curve

The subject of this problem is Heighway Dragon~!
https://projecteuler.net/problem=220

Let D₀ be the two-letter string "Fa". For n≥1, derive D_n from D_n-1 by the string-rewriting rules:

"a" → "aRbFR"
"b" → "LFaLb"

Thus, D₀ = "Fa", D₁ = "FaRbFR", D₂ = "FaRbFRRLFaLbFR", and so on.

These strings can be interpreted as instructions to a computer graphics program, with "F" meaning "draw forward one unit", "L" meaning "turn left 90 degrees", "R" meaning "turn right 90 degrees", and "a" and "b" being ignored. The initial position of the computer cursor is (0,0), pointing up towards (0,1).

Then D_n is an exotic drawing known as the Heighway Dragon of order n. For example, D₁₀ is shown below; counting each "F" as one step, the highlighted spot at (18,16) is the position reached after 500 steps.

What is the position of the cursor after 10¹² steps in D₅₀ ?
Give your answer in the form x,y with no spaces.

Approaches

1. make this way using by phyton and turtle.
2. think more deeply to solve it.
3. I have to find pattern or math skill.

this is my phython coding.

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__author__ = 'ddavid' import turtle a = 'aRbFR' b = 'LFaLb' s = 'Fa' wn = turtle.Screen() # Creates a playground for turtles tt = turtle.Turtle() # Create a turtle, assign to alex tt.speed(0) tt.left(90) tt.left(90) for i in range(14): s = s.replace('a', 't') s = s.replace('b', 'p') s = s.replace('t', a) s = s.replace('p', b) print(s) print('\n') if i == 9: for d in range(len(s)): print(s[d-1]) if s[d-1] == 'F': tt.forward(2) elif s[d-1] == 'L': tt.left(90) elif s[d-1] == 'R': tt.right(90) wn.mainloop() # Wait for user to close window

if you run this code
you will see this result.

Have Fun~ enyoing euler problem site.

reference
https://en.wikipedia.org/wiki/Dragon_curve