Let $x, y,z$ satisfy the following inequalities $\begin{cases} | x + 2y - 3z| \le 6 \\
| x - 2y + 3z|  \le  6  \\
| x - 2y - 3z|  \le 6 \\
| x + 2y + 3z|  \le  6 \end{cases}$
Determine the greatest value of $M = |x| + |y| + |z|$.
To determine the greatest value of \( M = |x| + |y| + |z| \) given the inequalities:
\[
\begin{cases} 
| x + 2y - 3z| \le 6 \\
| x - 2y + 3z| \le 6 \\
| x - 2y - 3z| \le 6 \\
| x + 2y + 3z| \le 6 
\end{cases}
\]

1. **Understanding the inequalities**:
   Each inequality represents a region in 3-dimensional space. The absolute value inequalities can be interpreted as planes that bound a region. For example, \( |x + 2y - 3z| \le 6 \) represents the region between the planes \( x + 2y - 3z = 6 \) and \( x + 2y - 3z = -6 \).

2. **Identifying the vertices**:
   The solution to the system of inequalities is the intersection of these regions, which forms a convex polyhedron. The vertices of this polyhedron can be found by solving the system of equations formed by setting each inequality to equality. The vertices are:
   \[
   \{(\pm 6,0,0), (0,\pm 3,0), (0,0,\pm 2)\}
   \]

3. **Analyzing the function \( M = |x| + |y| + |z| \)**:
   For any fixed \( M > 0 \), the graph of \( M = |x| + |y| + |z| \) is a plane in the first octant. The goal is to find the maximum value of \( M \) such that the plane \( M = |x| + |y| + |z| \) intersects the convex polyhedron formed by the inequalities.

4. **Comparing the convex hulls**:
   The convex hull of the vertices \(\{(\pm 6,0,0), (0,\pm 3,0), (0,0,\pm 2)\}\) is a polyhedron centered at the origin. The graph of \( M = |x| + |y| + |z| \) is a plane that intersects this polyhedron. Since the polyhedron is symmetric and centered at the origin, the maximum value of \( M \) occurs when the plane \( M = |x| + |y| + |z| \) just touches the farthest vertex of the polyhedron.

5. **Determining the maximum \( M \)**:
   The farthest vertex from the origin in the set \(\{(\pm 6,0,0), (0,\pm 3,0), (0,0,\pm 2)\}\) is \((6,0,0)\). At this vertex, \( M = |6| + |0| + |0| = 6 \).

Therefore, the greatest value of \( M = |x| + |y| + |z| \) is \( 6 \).

The final answer is \(\boxed{6}\).