
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
Ellipse $E$: $\dfrac{x^{2}}{a^{2}}+ \dfrac{y^{2}}{b^{2}}=1(a > b > 0)$ passes through point $A(0, -1)$, and its eccentricity is $\dfrac{\sqrt{2}}{2}$.

$(1)$ Find the equation of ellipse $E$.

$(2)$ A line passing through point $(1, 1)$ with slope $k$ intersects ellipse $E$ at two distinct points $P$ and $Q$ (both different from point $A$). Determine whether the sum of the slopes of lines $AP$ and $AQ$ is a constant. If it is a constant, find its value; if not, explain why.

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