\(\begin{align}
& M\ge \left| a+b+c+1 \right| \\
& M\ge \left| a-b+c-1 \right| \\
& M\ge \left| \frac{1}{4}a+\frac{1}{2}b+c+\frac{1}{8} \right| \\
& M\ge \left| \frac{1}{4}a-\frac{1}{2}b+c-\frac{1}{8} \right| \\
& \\
& M+M+2M+2M \\
& \ge \left| a+b+c+1 \right|+\left| a-b+c-1 \right|+\left| \frac{1}{2}a-b+2c-\frac{1}{4} \right|+\left| \frac{1}{2}a+b+2c+\frac{1}{4} \right| \\
& \ge \left| a+b+c+1-\left( a-b+c-1 \right)+\frac{1}{2}a-b+2c-\frac{1}{4}-\left( \frac{1}{2}a+b+2c+\frac{1}{4} \right) \right| \\
& =\frac{3}{2} \\
& \\
& M\ge \frac{1}{4} \\
\end{align}\)
