CONO-SUR 2018 Problem 2

Prove that every positive integer can be formed by the sums of powers of 3, 4 and 7, where do not appear two powers of the same number and with the same exponent. Example: $2= 7^0 + 7^0$ and $22=3^2 + 3^2+4^1$ are not valid representations, but $2=3^0+7^0$ and $22=3^2+3^0+4^1+4^0+7^1$ are valid representations.