
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 F<sub>1</sub> and F<sub>2</sub> are the left and right foci respectively of the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ (where $a > b > 0$) with semi-focal length $c$, the line $x = -\frac{a^2}{c}$ intersects the x-axis at point N, satisfying $\overrightarrow{F_{1}F_{2}} = 2\overrightarrow{NF_{1}}$, and $|\overrightarrow{F_{1}F_{2}}| = 2$. Let A and B be two points on the upper half of the ellipse such that $\overrightarrow{NA}=\lambda \overrightarrow{NB}$, where $\lambda \in [\frac{1}{5}, \frac{1}{3}]$.
(1) Find the equation of the ellipse and the range of values for the slope $k$ of line AB;
(2) Tangent lines to the ellipse are drawn at points A and B, intersecting at point P. Determine whether point P always lies on a fixed line, explaining the reason.

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