Find $a$ and $b$ such that $7^7|(a+b)^7-a^7-b^7$

Question

Find a pair of positive integers $(a,b)$ such that $7$ does not divide $ab(a+b)$ and $7^7$ divides $(a+b)^7-a^7-b^7$

$$(a+b)^7-a^7- b^7=7ab(a+b)(a^2+ab+b^2)^2$$

By application of binomial and completing the square I reduced the problem to find $(a,b)$ such that $7^3$ divides $a^2+ab+b^2$

How to proceed further?

Answer 1

By inspection the couple $(a,b)=(18,1)$ works. If you fix $b=1$, the problem becomes simpler.