
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 $F_1$ and $F_2$ be the left and right foci of the ellipse $\frac{x^2}{4}+y^2=1$ respectively.

$(1)$ If $P$ is a point on the ellipse in the first quadrant, and $\overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = -\frac{5}{4}$, find the coordinates of point $P$;

$(2)$ Suppose a line $l$ passing through the fixed point $M(0,2)$ intersects the ellipse at two distinct points $A$ and $B$, and $\angle AOB$ is acute (where $O$ is the origin), find the range of values for the slope $k$ of line $l$.

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