
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 \( S \) be a set consisting of 100 elements from the natural numbers not exceeding 10,000. Consider all points in space where each coordinate belongs to the set \( S \). Each of the resulting 1,000,000 points \((x, y, z)\) is assigned a balloon with a number written on it given by \(\frac{x^{2}+y^{2}+z^{2}}{xy+yz+zx}\). What is the maximum number of balloons that can have the number 2 written on them?

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