
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
\( m \) and \( n \) are two different positive integers. Find the common complex root of the equations
\[
\begin{aligned}
x^{m+1} - x^{n} + 1 = 0 \\
x^{n+1} - x^{m} + 1 = 0
\end{aligned}
\]

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