If $a^4 + a^3 + a^2 + a + 1 = 0$, find the value of $a^{2m} + a^m + 1/a^m + 1/a^{2m}$ when $m$ is a multiple of 5, and find $a^{4m} + a^{3m} + a^{2m} + a^m$.
$a^4 + a^3 + a^2 + a + 1 = 0 \Rightarrow a^5 - 1 = 0$, hence $a$ is the fifth root of unity $a = \left(\cos\frac{2K\pi}{5} + i\sin\frac{2K\pi}{5}\right)$. $a^{2m} + a^m + 1/a^m + 1/a^{2m} = 5$ if $m$ is a multiple of 5. $a^{4m} + a^{3m} + a^{2m} + a^m = -1$.
If $a^4 + a^3 + a^2 + a + 1 = 0$, find the value of $a^{2m} + a^m + 1/a^m + 1/a^{2m}$ when $m$ is a multiple of 5, and find $a^{4m} + a^{3m} + a^{2m} + a^m$.