
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 \( n \geq 4 \) be an integer. Find all positive real solutions to the following system of \( 2n \) equations:
$$
\begin{aligned}
a_{1} &= \frac{1}{a_{2n}} + \frac{1}{a_{2}}, & a_{2} &= a_{1} + a_{3}, \\
a_{3} &= \frac{1}{a_{2}} + \frac{1}{a_{4}}, & a_{4} &= a_{3} + a_{5}, \\
a_{5} &= \frac{1}{a_{4}} + \frac{1}{a_{6}}, & a_{6} &= a_{5} + a_{7}, \\
&\vdots & &\vdots \\
a_{2n-1} &= \frac{1}{a_{2n-2}} + \frac{1}{a_{2n}}, & a_{2n} &= a_{2n-1} + a_{1}.
\end{aligned}
$$

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