
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
What conditions must the integers \( m, n, p, q \) satisfy so that the set of four numbers

\[ 
\begin{aligned}
x &= 2 m p + 2 n q, \\
y &= |2 m q - 2 n p|, \\
z &= |m^2 + n^2 - p^2 - q^2|, \\
u &= m^2 + n^2 + p^2 + q^2 
\end{aligned}
\]

forms a quadruplet (a spatial Pythagorean quadruplet) such that they do not have a common divisor greater than 1?

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