
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 ellipse \( E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 \) with \( a > b > 0 \), its minor axis length is 4 and the eccentricity is \( e_{1} \). The hyperbola \( \frac{x^{2}}{m}-\frac{y^{2}}{n}=1 \) with \( m > 0 \) and \( n > 0 \) has asymptotes \( y= \pm x \) and an eccentricity \( e_{2} \), and \( e_{1} \cdot e_{2}=1 \).

(1) Find the equation of the ellipse \( E \).

(2) Let the right focus of ellipse \( E \) be \( F \). A line passing through the point \( G(4,0) \) with a non-zero slope intersects ellipse \( E \) at points \( M \) and \( N \). Let the slopes of lines \( FM \) and \( FN \) be \( k_{1} \) and \( k_{2} \), respectively. Determine if \( k_{1} + k_{2} \) is a constant value. If it is, find this constant value; if it is not, explain why.

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