
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 the vertices and foci of the hyperbola $x^{2}- \frac{y^{2}}{2} =1$ are respectively the foci and vertices of the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$.

(Ⅰ) Find the equation of the ellipse $C$;

(Ⅱ) It is known that a line $l$ passes through the right focus $F_{2}$ of the ellipse $C$, intersecting the ellipse at points $A$ and $B$. When the line $l$ is not perpendicular to either coordinate axis, is there always a point $P$ on the $x$-axis such that the angles of inclination of lines $PA$ and $PB$ are complementary? If such a point exists, find the coordinates of $P$; if not, explain why.

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