
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
The equation \( x^{2} + 5x + 1 = 0 \) has roots \( x_{1} \) and \( x_{2} \). Find the value of the expression

$$
\left(\frac{x_{1} \sqrt{6}}{1 + x_{2}}\right)^{2} + \left(\frac{x_{2} \sqrt{6}}{1 + x_{1}}\right)^{2}
$$

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