A naïve solution would iterate over positive integers and check their luck until the ~K^\text{th}~ lucky integer is found. Such a solution is worth ~20\%~ of points.

A better solution is possible if a pattern is noticed in lucky numbers.

Specifically, if we substitute the digit ~4~ with ~0~, and ~7~ with ~1~, the lucky numbers correspond to binary number notation, with an additional quirk that leading zeros are allowed.

A solution iteratively generating the first ~K~ lucky numbers using this observation has a complexity of ~\mathcal O(K)~ and is worth ~50\%~ of points.

Another speedup can be made by observing the number of lucky numbers with a given length. There are ~2^N~ lucky numbers with length ~N~ (since each of ~N~ positions can contain one of two digits).

Knowing this, we can determine the length ~L~ of the required lucky number and the index ~P~ of that number among all lucky numbers of length ~L~. Then, it is sufficient to output the number ~P-1~ in binary with the appropriate number of leading zeros, substituting digits ~0~ and ~1~ with ~4~ and ~7~, respectively.