Automata theory. Problem #$2.$ Let $L=\{xin(a+b{)}^{**}|$#${?}_a(x)=$#${?}_b(x)\}$

Show that if xinL and $x$ begins and ends with the same symbol, then $x=yz$ for some nonempty strings $y,$zinL. $($Hint$.$ For $x=x_{1}dots{x}_n,x_1,$dots,$x_{n}in\{a,b\},$ define the function $f$:$\{0,$dots,$n\}Z$ by letting $f(i)=$#${?}_a(x_{1}dots{x}_i)-$#${?}_b(x_{1}dots{x}_i),$ and argue that $y$ and $z$ exist.$)$

Use $(1)$ to find a CFG generating $L$ in which $S$ is the only nonterminal.