
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 any \( n > 2 \):

\[
\begin{aligned}
& a_{11} x_{1} + a_{12} x_{2} + \ldots + a_{1n} x_{n} = 0 \\
& a_{21} x_{1} + a_{22} x_{2} + \ldots + a_{2n} x_{n} = 0 \\
& \vdots \\
& a_{n1} x_{1} + a_{n2} x_{2} + \ldots + a_{nn} x_{n} = 0
\end{aligned}
\]

given that:

a) All coefficients \( a_{ij} \) are positive.

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

c) \( a_{11} = a_{22} = \ldots = a_{nn} = \frac{1}{2} \).

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