
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
Given \( x, y, z \in \mathbb{Z}_{+} \) with \( x \leq y \leq z \), find the number of sets of solutions that satisfy the equation \(\frac{1}{x}+\frac{1}{y}+\frac{1}{z}=\frac{1}{2}\).

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