
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
Given the hyperbola \( C: \frac{x^2}{4} - y^2 = 1 \). The line \( l \) passing through point \( M(1, -1) \) intersects the right branch of the hyperbola at points \( A \) and \( B \), and intersects the x-axis at point \( N \). Suppose
\[
\overrightarrow{MA} = \lambda_1 \overrightarrow{AN}, \quad \overrightarrow{MB} = \lambda_2 \overrightarrow{BN} \quad (\lambda_1, \lambda_2 \in \mathbb{R}).
\]
Find the range of the value of \(\frac{\lambda_1}{\lambda_2} + \frac{\lambda_2}{\lambda_1}\).

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