
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
Considering the remainder when divided by 3, we get that \( x^{2} + y^{2} + z^{2} \) gives a remainder of 1. Since the square of a natural number can only give a remainder of 0 or 1, we see that two of the three numbers must be divisible by 3, meaning they are equal to 3. If these are the first two, then \( z = 2 \). If \( z = 3 \), there is no solution.

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