
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) with the equation $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 (a > b > 0)$, its eccentricity $e = \frac{\sqrt{2}}{2}$, and its left and right foci $F_1$ and $F_2$, respectively. Point $P(2, \sqrt{3})$ satisfies the condition that $F_2$ lies on the perpendicular bisector of line segment $PF_1$.

1. Find the equation of ellipse (C).
2. If a non-zero sloped line $l$ intersects with the x-axis and ellipse (C) consecutively at points $A(2,0)$, $M$, and $N$, and $\angle NF_2F_1 = \angle MF_2A$, find the range of values for the slope $k$.

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