
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
If the distinct non-zero numbers $x + (y - z), y + (z - x), z + (x - y)$ form an arithmetic progression with a common difference of $d$, find the relationship between the common difference $d$ and the variables $x, y, z$.
**A)** $x = y - d, z = y + \frac{d}{2}$  
**B)** $x = y + d, z = x + \frac{d}{2}$  
**C)** $x = y + \frac{d}{2}, z = y + d$  
**D)** $x = z - \frac{d}{2}, y = z - d$

Present the answer in LaTex format: \boxed{Your answer}
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