
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  $x,y,z$  be complex numbers satisfying \begin{align*}
z^2 + 5x &= 10z  
y^2 + 5z &= 10y  
x^2 + 5y &= 10x
\end{align*}
Find the sum of all possible values of  $z$ .

*Proposed by Aaron Lin*

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