
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

$$
\begin{aligned}
& a_{11} x+a_{12} y+a_{13} z=0 \\
& a_{21} x+a_{22} y+a_{23} z=0 \\
& a_{31} x+a_{32} y+a_{33} z=0
\end{aligned}
$$

given that the coefficients satisfy the following conditions:

a) All coefficients are positive,

b) The sum of the coefficients in each row and each column is 1,

c) $a_{11}=a_{22}=a_{33}=1/2$.

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