I've been trying for hours to figure out, how to form a CFG describing this language $L$: $$L=\{ w\in\{a,b\}^* \mid w\text{ is of the form }a^nxb^{n+2}\text{, where }x\text{ is a string of length }3\text{ in }\{a,b\}^* \}.$$

The problem I'm having is related to the first $n$ of $a$'s. I can't wrap my head around how can this be described properly, since the $x$ can also contain 3 $a$'s.

This is my current solution. It does not work because it doesn't take the $n$ $a$'s to the front.

A -> aB | bB
B -> aC | bC
C -> aD | bD
D -> bE
E -> bF
F -> bF | _ 

$S \to aSb \mid Abb, A \to aaa \mid aab \mid \ldots \mid bbb.$ That's it. First create an equal number of $a$s and $b$s with $S$. Eventually go to the $Abb$ rule creating two extra $b$s and then terminate with $A$ to any string of length $3$.

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