25.rb

def solve(key)
  (0..).inject(1) do |memo, n|
    return n if memo == key
    (memo * 7) % 20201227
  end
end

def transform(subject, n)
  (1..n).inject(1) { |memo, _| (memo * subject) % 20201227 }
end

data = [ 5764801, 17807724 ]
ncard, ndoor = solve(card), solve(door)
p [ ncard, ndoor ]
p [ transform(door, ncard), transform(card, ndoor) ]