Martian Inequality !

If $a$ , $b$ , and $c$ are three positive real numbers then the minimum value of

$\large\ \frac { a + 3c }{ a + 2b + c } + \frac { 4b }{ a + b + 2c } - \frac { 8c }{ a + b + 3c }$ .

is $\alpha + \beta \sqrt { 2 }$ , where $\alpha ,\beta \in \mathbb Z$ . Find the value of $| \alpha + \beta|$ .

Notation: $|\cdot |$ denotes the absolute value function .