
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 $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ $(a > b > 0)$, which has an eccentricity of $\frac{\sqrt{2}}{2}$, a line $l$, passing through its left focus $F(-1,0)$ and not coinciding with the $x$-axis, intersects the ellipse at points $M$ and $N$.
1. Find the equation of the ellipse $C$.
2. Is there a fixed point $Q$ on the $x$-axis such that the distances from point $F$ to the lines $QM$ and $QN$ are always equal? If it exists, find the coordinates of point $Q$; if not, explain why.

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