
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 right focus of the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ is $F(1,0)$, and the point $A(2,0)$ is on the ellipse. The line $l$ passing through $F(1,0)$ intersects the ellipse $C$ at two distinct points $M$ and $N$.
(1) Find the equation of the ellipse $C$;
(2) If the slope of line $l$ is $1$, find the length of the line segment $MN$;
(3) If the perpendicular bisector of line segment $MN$ intersects the $y$-axis at point $P(0,y_{0})$, find the range of $y_{0}$.

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