
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 an ellipse $C$: $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$ with the left and right foci $F_{1}$ and $F_{2}$, respectively. If the point $N(0, \sqrt{3})$ lies on the ellipse and $\triangle NF_{1}F_{2}$ is an equilateral triangle.<br/>$(1)$ Find the standard equation of the ellipse $C$.<br/>$(2)$ A line $l$ passing through $F_{1}$ with slope $k$ intersects the ellipse $C$ at points $A$ and $B$. If $\angle AF_{2}B$ is obtuse, find the range of values for $k$.

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