
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
Solve the system of equations for \( x, y, z \in \mathbb{R}^{+} \):
$$
\left\{\begin{array}{l}
x^{2}+y^{2}+\sqrt{3} x y=20+8 \sqrt{3} \\
y^{2}+z^{2}=13 \\
z^{2}+x^{2}+x z=37
\end{array}\right.
$$

Find \((x, y, z)\).

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