Find a CFG for the language $\{ x\$y \mid x,y\in\{a,b\}^* \wedge |x| \ne |y| \}$? Consider the language below, on the alphabet $\Sigma = \{a,b,\$\}$:
$$L = \left\{ x$y \mid x,y\in\{a,b\}^* \land \left|x\right| \ne \left|y\right| \right\}$$

I need to define a CFG for this language. I've tried couple of CFGs but they all failed in one way or another.

I'd be glad for help.
Hint:
$$
L = \{ \Sigma^{m+1} \Sigma^n \$ \Sigma^n : n,m \geq 0 \} \cup \{ \Sigma^n \$ \Sigma^n \Sigma^{m+1} \},
$$
where $\Sigma = \{a,b\}$.