
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
Raise both equations to the square and sum them up. We obtain the following system: \( (x^2 + t^2)(z^2 + y^2) = 50 \). Since \( x, y, z, t \) are natural numbers, this equality is possible only in the following cases:
\[
\begin{cases}
x^2 + t^2 = 5 \\
z^2 + y^2 = 10 \\
\end{cases}
\quad
\begin{cases}
x^2 + t^2 = 10 \\
z^2 + y^2 = 5 \\
\end{cases}
\quad
\begin{cases}
x^2 + t^2 = 25 \\
z^2 + y^2 = 2 \\
\end{cases}
\quad
\begin{cases}
x^2 + t^2 = 2 \\
z^2 + y^2 = 25 \\
\end{cases}
\]

From the first system, we find that either \( x = 2, t = 1 \) or \( x = 1, t = 2; z = 3, y = 1 \) or \( z = 1, y = 3 \). Then, we get four solutions: 
\[ (2, 1, 1, 3), (1, 1, 2, 3), (1, 3, 2, 1), (2, 3, 1, 1) \]
Only one, \( (1,1,2,3) \) satisfies the original system.

From the second system, we find that \( x = 3, t = 1; x = 1, t = 3; z = 2, y = 1; y = 2, z = 1 \). Then, this system's solution satisfies the initial system resulting in:
\[ (3, 2, 1, 1) \]

Similarly, we find two more solutions: 
\[ (4, 1, 3, 1), (1, 3, 4, 1) \]

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