
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
A line passing through the origin intersects the ellipse $(C)$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ at points $A$ and $B$. Point $P$ is a point on the ellipse distinct from $A$ and $B$. The slopes of lines $PA$ and $PB$ exist and their product is $- \frac{1}{4}$.

1. Find the eccentricity of the ellipse $(C)$.
2. Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse, respectively. A line $(l)$ with slope $k$ passes through the right focus and intersects the ellipse at points $M$ and $N$. If point $F_{1}$ is inside the circle with diameter $|MN|$, find the range of $k$.

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