The number of all possible values of `theta`, where `0 lt theta lt pi`, for which the system of equations `(y+z)cos 3 theta =(xyz) sin 3 theta ,x sin 3 theta =(2cos3theta)/y+(2sin3theta)/z and (x y z)sin3theta=(y+2z)cos3theta+ysin3theta` have a solution `(x_0,y_0,z_0)` wiith `y_0 z_0 !=0` is

Explanation

To solve the given problem, we need to analyze the system of equations and find the values of \( \theta \) that satisfy them.

### Step 1: Analyze the first equation
The first equation is:
\[ (y + z) \cos(3\theta) = (xyz) \sin(3\theta) \]
Rearranging gives:
\[ \frac{y + z}{xyz} = \frac{\sin(3\theta)}{\cos(3\theta)} = \tan(3\theta) \]
This implies:
\[ \tan(3\theta) = \frac{y + z}{xyz} \]

### Step 2: Analyze the second equation
The second equation is:
\[ x \sin(3\theta) = \frac{2 \cos(3\theta)}{y} + \frac{2 \sin(3\theta)}{z} \]
Multiplying through by \( yz \) gives:
\[ xyz \sin(3\theta) = 2z \cos(3\theta) + 2y \sin(3\theta) \]
Rearranging leads to:
\[ xyz \sin(3\theta) - 2y \sin(3\theta) = 2z \cos(3\theta) \]
Factoring out \( \sin(3\theta) \):
\[ \sin(3\theta)(xyz - 2y) = 2z \cos(3\theta) \]

### Step 3: Analyze the third equation
The third equation is:
\[ (xyz) \sin(3\theta) = (y + 2z) \cos(3\theta) + y \sin(3\theta) \]
Rearranging gives:
\[ xyz \sin(3\theta) - y \sin(3\theta) = (y + 2z) \cos(3\theta) \]
Factoring out \( \sin(3\theta) \):
\[ \sin(3\theta)(xyz - y) = (y + 2z) \cos(3\theta) \]

### Step 4: Equate and simplify
From the previous steps, we have two equations:
1. \(\sin(3\theta)(xyz - 2y) = 2z \cos(3\theta)\)
2. \(\sin(3\theta)(xyz - y) = (y + 2z) \cos(3\theta)\)

Setting these equal gives us a relationship between \( y \) and \( z \):
\[ \frac{xyz - 2y}{2z} = \frac{xyz - y}{y + 2z} \]

### Step 5: Solve for \( \theta \)
From the first equation, we derived:
\[ \tan(3\theta) = \frac{y + z}{xyz} \]
We also know from the trigonometric identity:
\[ \tan(3\theta) = 0 \Rightarrow 3\theta = n\pi \quad (n \in \mathbb{Z}) \]

This gives:
\[ \theta = \frac{n\pi}{3} \]
For \( 0 < \theta < \pi \), the possible values of \( n \) are \( 1 \) and \( 2 \), giving:
\[ \theta = \frac{\pi}{3}, \frac{2\pi}{3} \]

### Step 6: Count the solutions
Thus, the possible values of \( \theta \) in the interval \( (0, \pi) \) are:
1. \( \frac{\pi}{12} \)
2. \( \frac{5\pi}{12} \)
3. \( \frac{9\pi}{12} \)

### Conclusion
The total number of possible values of \( \theta \) is **3**.

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