
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 the ellipse \( E: \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a>b>0) \) have a minor axis of length 4 and eccentricity \( e_{1} \). Consider the hyperbola \(\frac{x^{2}}{m}-\frac{y^{2}}{n}=1 (m>0, n>0) \) with asymptotes \( y=\pm x \) and eccentricity \( e_{2} \). Given that \( e_{1} e_{2}=1 \):

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

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