
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 a hyperbola \( C: \frac{x^{2}}{4} - y^{2} = 1 \), a line \( l \) passing through the point \( M(1, -1) \) intersects the right branch of the hyperbola at points \( A \) and \( B \) and intersects the \( x \)-axis at point \( N \). Assume
\[ 
\overrightarrow{M A} = \lambda_{1} \overrightarrow{A N}, \quad \overrightarrow{M B} = \lambda_{2} \overrightarrow{B N} \quad (\lambda_{1}, \lambda_{2} \in \mathbf{R}).
\]

Determine the range of values of \(\frac{\lambda_{1}}{\lambda_{2}} + \frac{\lambda_{2}}{\lambda_{1}}\).

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