
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
Let $A(x_{1},x_{2})$ and $B(y_{1},y_{2})$ be two points located above the $x$-axis on the ellipse $E$: $\dfrac{x^{2}}{9}+y^{2}=1$, and $x_{1}+x_{2}=2$.
$(1)$ If $y_{1}+y_{2}=1$, find the equation of the perpendicular bisector of segment $AB$;
$(2)$ Find the minimum value of the $y$-intercept of line $AB$.

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