
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 the following system of equations be satisfied for positive numbers \(x, y, z\):

\[ 
\left\{
\begin{array}{l}
x^{2} + x y + y^{2} = 27 \\
y^{2} + y z + z^{2} = 25 \\
z^{2} + x z + x^{2} = 52
\end{array}
\right.
\]

Find the value of the expression \(x y + y z + x z\).

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