A nice equation $!!!$

$\frac{a_2}{2!}+\frac{a_3}{3!}+\frac{a_4}{4!}+\frac{a_5}{5!}+\frac{a_6}{6!}+\frac{a_7}{7!}=\frac{5}{7}$

Given the above, where $0\leq a_k\leq k$ and $a_n$ is an integer, find minimum value of $a_2+a_3+a_4+a_5+a_6+a_7$ .

Notation: $!$ denotes the factorial notation , For example: $8! =1 \times 2 \times 3 \times 4 \times 5 \times 6 \times 7 \times 8$ .

Request : please contribute to this