IMAS 2021–2022 Junior Division, Round 2 Question 6

We define: $\boxed2=1\times2\times3$, $\boxed3=2\times3\times4$, $\boxed4=3\times4\times5$, $\boxed5=4\times5\times6$, …. If $\frac1{\boxed6}-\frac1{\boxed7}=\frac1{\boxed7}\times A$, what is the value of $A$?