Real parts, imaginary parts, moduli and arguments

Let $z=\dfrac{2+i}{3-i}$. Without using a calculator, find the real and imaginary parts of

(a) $\dfrac{1+z}{1-z}$,

(b) $z+\dfrac1z$.

Hence, find the modulus and argument of $\dfrac{1+z}{1-z}$ and $z+\dfrac1z$.