
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 $C: \frac{x^{2}}{2}+y^{2}=1$ with its left focus $F$, a line $l$ that is not perpendicular to the $x$-axis and does not pass through point $F$ intersects the ellipse $C$ at points $A$ and $B$.
(1) If the sum of the slopes of lines $FA$ and $FB$ is $0$, does the moving line $l$ necessarily pass through a fixed point? If so, find the coordinates of this fixed point; if not, explain the reason.
(2) If $FA \perp FB$, and the distance from the origin to line $l$ is $d$, find the range of values for $d$.

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