
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 \( p = 4n + 1 \) be a prime, where \( p = x^2 + y^2 \) for \( x, y \in \mathbb{Z}_+ \). Then there exists a geometric progression of \( 2n \) positive integers \( a_1, a_2, \ldots, a_{2n} \) such that the \( 4n \) positive integers
\[ x^2 a_1, x^2 a_2, \ldots, x^2 a_{2n}, y^2 a_1, y^2 a_2, \ldots, y^2 a_{2n} \]
form a complete residue system modulo \( p \).

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