
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 that $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C$: $\dfrac{x^{2}}{a^{2}} + \dfrac{y^{2}}{b^{2}} = 1 (a > b > 0)$, and the eccentricity is $\dfrac{\sqrt{2}}{2}$, point $A(-\dfrac{\sqrt{2}}{2}, \dfrac{\sqrt{3}}{2})$ is on the ellipse $C$.
1. Find the equation of the ellipse $C$.
2. Determine whether there exists a line $l$ with slope $k$ that intersects the ellipse $C$ at two distinct points $M$ and $N$, such that the angles of inclination of lines $F_{2}M$ and $F_{2}N$ are complementary, and whether line $l$ always passes through a fixed point. If such a point exists, find its coordinates; if not, explain why.

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