
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
Let  $x$ ,  $y$ , and  $z$  be nonnegative integers that are less than or equal to 100.  Suppose that  $x + y + z$ ,  $xy + z$ ,  $x + yz$ , and  $xyz$  are (in some order) four consecutive terms of an arithmetic sequence.  Compute the number of such ordered triples  $(x, y, z)$ . 


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