
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
Consider the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with foci $F_{1}(-c,0)$, $F_{2}(c,0)$. The upper vertex $M$ of the ellipse satisfies $\overrightarrow{F_{1}M} \cdot \overrightarrow{F_{2}M}=0$.
(I) Find the eccentricity $e$ of the ellipse $C$;
(II) Let $N(0,2)$ be the center of a circle that intersects the ellipse $C$. The maximum radius of such a circle is $\sqrt{26}$.
(i) Find the equation of the ellipse $C$ in this case;
(ii) Determine whether there exist two points $A$, $B$ on the ellipse $C$ that are symmetric with respect to the line $l$: $y=kx-1 (k \neq 0)$. If they exist, find the range of $k$; otherwise, explain the reason.

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