Give context$-$free grammars that generate the following languages over the alphabet $=\{a,b\}.$

In each case, try to minimize the number of variables and specify all components of the quadruple

$(V,,R,S).$

The full language $($Optimal solution has only one variable$)$ and $w$ begins and ends with the same symbol $).$

The full language $\backslash$Sigma $*$

$\{$a$^$mb$^$n mn$\}($Optimal solution has only one variable$)$

$\{$w $|$ w contains at least four a$\text{'}$s$\}$