
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 the hyperbola $C$: $\frac{x^2}{2}-y^2=1$, where the coordinates of point $M$ are $\left(0,1\right)$.<br/>$(1)$ Let the line $l$ pass through point $M$ with a slope of $\frac{1}{2}$, intersecting the hyperbola $C$ at points $A$ and $B$. Find the length of segment $AB$;<br/>$(2)$ Let point $P$ be on the hyperbola $C$, and $Q$ be the point symmetric to point $P$ with respect to the $y$-axis. Let $k=\overrightarrow{MP}•\overrightarrow{MQ}$, find the range of values for $k$.

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