
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 an ellipse $C:\frac{{x}^{2}}{4}+\frac{{y}^{2}}{3}=1$, a line $l$ passing through the point $M(0,1)$ intersects the ellipse $C$ at points $A$ and $B$ (point $A$ is located above the $x$-axis). If $\overrightarrow{AM}=\frac{1}{2}\overrightarrow{MB}$, then the value of the slope $k$ of the line $l$ is ______.

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