
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 $C: \frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a>b>0)$ with an eccentricity of $e= \frac{\sqrt{2}}{2}$, a line $l$ passes through its right focus $F(1,0)$ and intersects the ellipse $C$ at points $A$ and $B$, without coinciding with the $x$-axis.

1. Find the equation of the ellipse $C$.
2. Does there exist a point $D$ on the $x$-axis such that the sum of the distances from point $O$ (the origin) to the lines $DA$ and $DB$ is constant? If so, find the coordinates of point $D$. If not, explain the reason.

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