
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 ordered quadruples $(x, y, z, w)$ of complex numbers satisfy
\begin{align*}
x + yz &= 8, \\
y + xw &= 11, \\
z + xy &= 11, \\
w + yz &= 5.
\end{align*} 
Find $x_1 + x_2 + \dots + x_n.$ where $(x_i, y_i, z_i, w_i)$ are the solutions.

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